Well it changes from President to President. Some use their predecessors a lot others not at all. There is a great book called the Presidents Club that explains the relationship between them all. I will attempt to give a very quick summary of them all

Truman- Really respected Hoover and used him as the chairman of one of his central committees. They worked together to transform the presidency which lead to them been named 3 and 5th most admired men in America in 1951.

Eisenhower- Thought Hoover was going senile and never used him. Did not like Truman and did not use him in any capacity. There was a slight thawing in the mood after the death of George Marshall but still no use in any capacity.

Kennedy- Did like Eisenhower thought he was an asshole but respected his Generalship. Used Eisenhower as a sounding board on foreign policy. Had an open invitation for him to consolt at any time. Used him a lot in the aftermath of the Bay of Pigs because of his large amount of expertise in that area. Offered various government positions to both Hoover and Truman such as the American Food for Peace program which they declined.

Johnson- Was the greatest user of past presidents called all former presidents in the hours after the assassination. Really liked and respected Eisenhower as he felt they both came from similar roots small town boys come good. Always asked Eisenhower for his opinion on all matters. Also greatly respected Truman as shown by the fact that the Medicare act was signed in Independence so that Truman could get the very first card.

Nixon- Well soon after his entry into office all his predecessors were dead so he did not use them at all.

Ford- Nixon was political poison and thus could not be used in any capacity.

Carter- Didn't really like Ford at the time, their relationship would only really begin when they realised they both hated Reagan on the plane ride to Sadat's Funeral. Plus again Nixon was still political poison and could not be used.

Reagan- Disliked Ford thought he was stupid and in over his head. That he was too conciliatory to be a good president. Really liked Nixon and used him a lot as both an advisor and as an envoy. Most notably with Nixons trip to Jedda after the Funeral of Sadat to meet with the Saudi's. Basically Nixon in Reagan's administration was that of the back-door fixer, problem solver. This suited both men fine as Nixon had a preference for Secrecy and Nixon was still problematic politically. Reagan always asked for Nixons advice and almost always took it, with one major exception of arms treaties.

Bush- Felt he had enough knowledge of foreign policy with his tenure at the UN, China and the CIA to not need Nixon's advice in this area. Thought Carter was a self-righteous, impatient perfectionist. However did realise that he had his uses and often tapped Carter and Ford to work together to solve problems like the 1989 Panama elections where Bush sent them both to observe the elections.

Clinton- Had the most former presidents to deal with. So I will be brief on each one. Reagan- Faded from view early in Clintons presidency as the alzheimer's took hold. Loved Bush considered him to be a father figure. There realtionship remains the closest of former presidents' Spoke at great lengths with Nixon and was very upset when he died. In 2011 Clinton said that to this day he still rereads a memo that Nixon sent him about the future of Russia and essentially a lot of Clintons policy in this area is Nixons. Used Carter and Ford to various length asking Ford in particular to help prevent the Republicans from impeaching him. Also used Carter as a peace envoy many times to both North Korea and Haiti.

Bush Jnr Couldn't really ask his father for help so didn't really speak to him. At first kept his distance from Clinton there were too many bad memories seeing his father lose to him in 1992. Bush called 1992 the worst year of his life. Used both his father and Clinton to run the Boxing Day Tsunami aid efforts. Again used Clinton and his father to help with Katrina. Grew to have a closer realtionship with Clinton as his presidency went on. This can be seen in the fact that Clinton was included in a photo of the Bush Clan taken after a Kennedy Centre benift.

Obama- Never had a great relationship with Bush as all the stuff said in the campaign and the first two years of office still piss of Bush. Bush did give some advice mostly on pardon policy as he had seen it take up a great deal of his own and Clintons presidency. Apart from that there is not as much info on Obama as he is the current president so the tail is not yet over.

In conclusion former Presidents are mostly used as unofficial advisers in foreign policy as that is the area were there is the most bipartisan agreement. Apart from that certain presidents use others in various ways from heavy political advisor to someone who can talk to the other side of the aisle. Once again if you would like to know more read The Presidents Club by Nancy Gibb and Michael Duffy.

Source Gibbs, N & Duffy, M The Presidents Club.

Conceptions of the Presidency were firmly rooted in 18th century Enlightenment and captured most prominently by Washington during his resignation from the army and later declining to run for a third term. Washington thus embodied the Republican notion serving while needed and then returning to your farm ( hence his comparisons with Cincinnatus). In fact one of the easiest ways to get elected to public office was to publicly declare you didn't want the office, public service was something you were called to do not sought after yourself. With all of this in mind it should be understood that it would not have been accepted Republican ideology to become deeply involved in future presidents administrations unless dire circumstances prevented otherwise, as such most of their involvement was very limited. You can read more about American conceptions of the Presidency in Ralph Ketcham's Presidents Above Party: The First American Presidency, 1789-1829

Washington and Adams had not enjoyed the greatest of relationships during the first presidency. Washington envisioned Adams role as being more legislative in nature (owing to the only real role of the VP in constitution as President of the Senate, and the senate swiftly forbade Adams from participating in debates). Washington thus excluded Adams from cabinet meetings and Adams was to proud to offer Washington advise. At the same time Washington was growing increasingly reliant on Alexander Hamilton, which would have dire ramifications during Adams presidency. When Adams called for the creation of a Federal army as the United States geared up for war against France, there was only one living man who could appear bipartisan and assume command, George Washington. Washington however was in to poor of health to assume command himself and actual command would largely be wielded by the second in command. Adams sought anyone but Hamilton (Knox etc..) however Washington was unwilling to accept anyone but Hamilton and without Washington's unquestioning moral superiority the army would be a very decisive issue. The political infighting between Hamilton and Adams would take a separate post but the TLDR is the army was the first of many political battles between the two men which would lead to the fracture of the Federalist party. John Quincy Adams would write

the army was the first decisive symptom of a schism in the Federal party itself, which accomplished its final overthrow and that of the administration."

Washington is of course dead by the time Jefferson is elected, although his private comments leave little doubt that he would not have approved of a Jefferson presidency.

I wish I could make a longer post on John Adams, but as far as I am aware he played little to no role in the Jeffersonian administrations. He was something of a pariah after the 1800 revolution, many Federalists didn't like him and even more Republicans didn't like him either. The one exception I can think of is when James Monroe went on his grand national tour, he broke bread with John Adams while in Massachusetts. It was from this symbolic event that a paper coined the term "Era of American nationalism" which has grown to refer to the post war of 1812 period until Jackson's election in 1828. Adams also criticized Federalists for their conduct during the war of 1812 and the embargoes, but I see little to no evidence this had any impact.

Jefferson, Madison and Monroe are an extremely interesting triumvirate the likes of which has never been seen since in American history. All three were Virginians, more than that they lived very close together (you can see Monticello from Monroe's home in the winter) enabling them to spend a great deal of time together. Most importantly all three were tightly interwoven allies in the political battles of the 1780's and 1790's. The relationship between Jefferson and each man was always stronger than the relationship between Monroe and Madison (indeed on at least two occasions the two men ran against each other for public office). Because of the closeness of their friendship and political battles it enabled each of them to take certain liberties that they probably would have otherwise not done. For instance Jefferson's insistent pleases that Monroe give up the Virginia governorship 1803 to go to France and purchase New Orleans. Because of their closeness it is difficult to overemphasize how important their opinion could be in shaping decisions of their friend-president.

Jefferson continued to play a large if limited role in the Madison administration. He offered advice to Madison on governing matters, sometimes it was arguably pretty bad especially in military affairs. For instance in the buildup to war he repeatedly sought to convince Madison to abandon plans for a conventional navy of Frigates and Ships of the Line in favor of his beloved gunboats. At the time the gunboat idea was considered a massive failure although some historians have issued reevaluations. Other examples include a plan to concentrate all of America's Frigates in one port, the flaw of the plan is quite obvious, but Madison wrote in a caring manner pointing out the flaws in the plan. The only major disagreement that Jefferson had with Madison's plans came with Madison's plans to charter a new national bank. While Jefferson had supported the Tariff of 1816 and the vastly strengthened navy he could never come on board with the idea of a national bank. Jefferson would work through proxies in Congress to attempt to block the bank, and floated ideas to Monroe for alternative solutions. Jefferson's most important contribution however was in bringing Monroe and Madison back together. The two men had been in competition for the nomination in the election of 1808, and hadn't rekindled their friendship. America was currently in the midst of diplomatic struggles between Britain and France and a Secretary of State that could be trusted was of the up most importance ( by contrast Adams terrible relationship with his own SOS had caused great harm to his presidency). I won't get into the details here but a few years into his presidency Madison needed to sack his secretary of state, Jefferson floated the idea of Monroe and brought the two men back together. Benjamin Rush reuniting Adams and Jefferson may get all the attention but bringing Madison and Monroe back together was of far more importance. Jefferson continued to give advice during Monroe's presidency on a variety of items, as well as sounding alarm at the increasing national measures of the federal government. His most important contribution probably came from his famous letter on the Missouri Crisis

but this momentous question, like a fire bell in the night, awakened and filled me with terror. I considered it at once as the knell of the Union. it is hushed indeed for the moment. but this is a reprieve only, not a final sentence

Jefferson's letter(intended for publication) had several intentions. One it was to add legitimacy to the Missouri Compromise, after the death of Washington, Jefferson was the founding father who stood largest in the minds of Americans. Secondly John Holmes ( to whom the letter had been written) was the only prominent Maine congressman who supported the compromise and it had cost him politically, having Jefferson give him support was an enormous boon. Adrienne Koch has the best work dealing with the correspondence between Madison and Jefferson although she oddly doesn't delve into the political aspects nearly as much as I would like but Jefferson and Madison: The Great Collaboration is an interesting read nonetheless, Madison and Jefferson by Burstein and Isenberg is an okay narrative account of the Madison and Jefferson's relationship. The authors attempt to recast Madison as the lead figure in the relationship, but they ultimately lose track of their subjects and seem to neglect some rather obvious pieces of history. Still the book is an okay read so long as the reader approaches it with an objective eye. Monroe and Jefferson's relationship has received far less attention with the only book dedicated to the subject being Noble Cunningham's Jefferson and Monroe. The book is however very short (more of an informative pamphlet really) and is very superficial, a very disappointing read considering Cunningham's reputation as one of the best if not the best political historians of the early Republic.

Madison plays much the same role in Monroe's presidency as Jefferson, offering advise on a multitude of issues. He was alarmed by the increase in tariffs in 1824, but as far as I am aware didn't act on it. Madison's largest post presidential contribution came during Andrew Jackson's presidency and the Nullification crisis. It had recently been discovered that Thomas Jefferson had authored the Kentucky resolution, where the term nullification was used. Jefferson had asked Madison to look after his legacy once he passed, and it now fell on Madison to defend Jefferson's legacy (although it's highly questionable if Madison actually defended Jefferson's original position). At the same time both nullifiers and Unionists wrote to Madison asking for political support in their struggle. Like Jefferson, Madison could be all things to all people and both sides had reason to hope for his support. Madison had after all authored the Virginia resolution, and had expressed great alarms at the tariffs of 1824 and 1828. At the same time however Madison had supported the tariff of 1816, threatened military action against Pennsylvania when they attempted to defy the Supreme Court, and opposed New England obstructions to government during the war of 1812. Whichever side Madison supported would gain an enormous asset in their battle for public and political opinion, by the 1830's Madison was the last national founding father still living. Ultimately despite his unease over the tariff of 1828 Madison came down hard on the nullifers writing

As this advice, if it ever see the light will not do it till I am no more it may be considered as issuing from the tomb where truth alone can be respected, and the happiness of man alone consulted. It will be entitled therefore to whatever weight can be derived from good intentions, and from the experience of one, who has served his Country in various stations through a period of forty years, who espoused in his youth and adhered through his life to the cause of its liberty, and who has borne a part in most of the great transactions which will constitute epochs of its destiny. The advice nearest to my heart and deepest in my convictions is that the Union of the States be cherished & perpetuated. Let the open enemy to it be regarded as a Pandora with her box opened; and the disguised one, as the Serpent creeping with his deadly wills into Paradise.

Henry Clay would wield Madison's declaration as a political tool against nullification. In regards to the emergence of two new National parties, Madison seems to have been on good terms with Jackson, Clay and Webster and not been overly involved in the political battles. In addition to some of the above books I would also recommend Susan Dunn's Dominion of Memories: Jefferson, Madison & the Decline of Virginia which focuses on the decline of Virginia and the ideology which Jefferson and Madison crafted that contributed to that decline, as you may surmise from the title she's fairly harsh on the two men. Drew McCoy does a great job picking up where most biographers leave off in The Last of the Fathers: James Madison & The Republican Legacy his sections dealing with slavery and nullification are especially useful.

James Monroe seems to have played a smaller role in the succeeding administrations, although it should be noted he only lived six years past the end of his presidency. He repeatedly refused to be drawn into partisan battles, partially because of his Republican ideology discussed in the first paragraph but also for financial reasons. Monroe was deep in dept after nearly 40 years of continuous service to America. Some of that debt was leftover from his time as ambassador to France and he sought repayment, hence his need to avoid political infighting. John Quincy Adams's supporters did attempt to have him run as VP for the 1828 election to no avail. A major political battle had erupted early in Jackson's presidency over his conduct in his invasion of Florida (during the Monroe presidency) and the incident's relation with then secretary of war John C. Calhoun. Again it would take too long to go into the details, but Jackson appears to have already been cold to Monroe because of the incident. Calhoun had made the short trip from Washington to Monroe's mansion outside of Richmond and begged for Monroe's support. Monroe again citing wanting to avoid political battles offered no help. It wasn't until Monroe lay deeply ill ( dying) and Jackson had directly attacked Monroe over the controversy (again to long to explain) that Monroe finally offered his version of events. Monroe has attracted no where near the scholarship as his role in America's formative years warrants so I can't cite much other than the before mentioned Cunningham book and Ammon's biography James Monroe: The Quest for National Identity.

Future Antebellum presidents are quite a bit more complicated, ideology had changed the idea of a party being a "good" had arrived. Moreover John Quincy Adams stayed quite active in politics as a member of the house meaning it would take an entire post just to describe his role. A whole slew of presidents died in office, and because of the politics of slavery few remained very influential.