I'd really like to get a better understanding of what it looks like when a political party goes away in America. What events led up to such drastic political shifts, what did the transitional process look like/how long did take, and how long did it take before the country returned to the 2-party system we've known and loved for so long?
I've been studying political biographies from the time of the Revolution to the 1850's. In that time period the status of the 'party' system can be summarized as:
In George Washington's first term the formal parties had not been established. Washington, and probably John Adams, did not want American politics to develop a party system; he envisioned a government of the best leaders working cooperatively, not in competition.
In George Washington's second term, against his best wishes, the first party system developed with Hamilton leading the Federalists and Jefferson/Madison leading the (old) Republicans. Washington sided with Hamilton on many key issues, leading the Republicans to become disenchanted with Washington (Jefferson later did not attend Washington's funeral probably due to bad blood between them).
Adams was nominally elected as a Federalist, even though A) he did not actively participate in the party system; and B) Hamilton (leader of the Federalists) actively tried to sabotage Adams' candidacy (twice). In fact, Hamilton had a lot to do with Adams losing to Jefferson during his reelection campaign.
With the election of Jefferson, the Federalist party decayed and eventually disappeared. No other Federalist was elected President. The last blow to the Federalists was when a group of Northeastern Federalists at the Hartford Convention plotted to secede from the Union during the War of 1812, immediately prior to the American victory (victory according to Madison and the Republicans that is), which discredited the Federalist party.
The first term of Monroe was a time with basically no party system in place (all the leading politicians were Republicans). Even John Quincy Adams, son of the only Federalist President, served as Secretary of State as a Republican, and eventual arch States Rights champion John C. Calhoun served as Secretary of War. However Monroe's second term was marred by vicious in-fighting among his cabinet who were positioning themselves to run for President after Monroe. So the equilibrium of having an effectively one-party system only lasted approximately one term (Monroe's first) before it broke down again into factions (much like what happened with Washington's Presidency).
Andrew Jackson rose up as a Presidential contender in 1824, getting the largest number of popular votes. However Jackson did not receive enough votes in the Electoral College to win; and J. Quincy Adams was voted in as President by the House of Representatives, backed by Speaker of the House Henry Clay. Basically this marked the beginning of the new two party system: the Democrats led by Jackson (and the true party leader Martin Van Buren) and the party that became known as the Whigs, led by Henry Clay. (Quincy Adams was, like his father, and anomaly and didn't fit in well with the defined parties; however he was allied with the Whigs and served as a strong anti-slavery Congressman for many years after his Presidency). Jackson of course won the Presidency in 1828, which has been characterized as the election that saw the rise in Direct Democracy by the People vs. the old-style Representative Democracy of the past.
The Whigs, as personified by Clay, championed an active Federal Government that supported American commerce and industry by implementing strong protectionist tariffs and sponsoring large public works projects (road and canal building, etc.) and a strong National Bank. The Democrats believed in a limited Federal government, having the States assume most powers, and when in power they eliminated the National Bank. This situation was in some ways reminiscent of the old Federalist/Republican two-party system: one party being pro-Federal Government and the other party wanting a small limited Federal government. (One Democratic President, James K. Polk, referred to the Whigs as 'Federals', which at that time was seen as a smear against them and was an allusion to the 'treason' of the Hartford Convention).
As we have seen, up until the early 1830's the American political system was more or less polarized along the axis of strong Federal Government vs. weak Federal Government. There had always been another undercurrent however, which was Slave states vs. Free Soil states, which was a sectional or geographic divide. George Washington had warned against the development of a strong sectional divide, and Thomas Jefferson saw the Missouri slavery controversy of 1820 as a "fire bell in the night". The leaders of both the Democratic and Whig parties for the most part were slaveholders (with the exception of J. Q. Adams and Van Buren), so slavery wasn't something that defined and divided the parties.
In the 1830's national politics became more and more sectional in nature. The initial leader of the sectional political orientation was then-Vice President John C. Calhoun and the political machine of South Carolina. When South Carolina threatened to 'nullify' certain Federal laws related to tariffs, President Andrew Jackson made a threat to lead a military invasion of the state (and nobody thought at the time that he was bluffing). But when the sectional issue of tariffs cooled off (due to a compromise engineered by Clay), the real sectional issue of Slave vs. Free-Soil states finally came to the forefront of national politics.
The Mexican American War became, after the predictable initial elation over the quick American victories, the embodiment of the Slave state vs. Free-soil state divide. Presidents Tyler and Polk openly viewed the annexation of Texas and the subsequent conquest of the territories of northern Mexico as a way to spread slavery westward as a means to counterbalance the Free-soil states coming into the Union in the North. Anti-slavery Whigs tried to pass legislation ('the Wilmot Proviso') that would have prevented slavery from being introduced into any territories that were captured from Mexico. This measure passed the Whig-dominated House of Representatives, but failed to pass in the Senate. And, to keep a long story from getting too much longer, that is where the fault lines of US politics inevitably shifted - from being about a strong Federal Government vs. a weak Federal Government to being about the North (Free-soil) vs. the South (Slavery) - a shift that took until about 1861 to complete.
The Whig party became untenable as it had been configured; and the Democratic party also was fractured. The old Whig leadership by the 1850's was dying out. The last Whig President to be elected, Mexican War hero General Zachary Taylor, was a slave owner who nevertheless supported the Wilmot Proviso and favored stopping the spread of slavery. One of Taylor's opponents in the 1848 election was former Democratic President Martin Van Buren, who was running as candidate of the Free-Soil Party instead of the Democratic party, with Charles Francis Adams - the son of Van Buren's former arch political enemy John Quincy Adams - as his running mate. The Democratic party candidate was firebrand pro-slavery advocate Lewis Cass. As it turned out, the Democratic party eventually re-formed after the Civil War and is alive and well today. The Whig party fell apart because the Northern Whigs (Free-soil) and the Southern Whigs (pro-Slavery) could no longer pull themselves together into a coherent party. The anti-slavery wing of the Whigs, along with some of the anti-slavery Democrats, eventually joined the new Republican party. In 1860 Abraham Lincoln was elected as the first (new) Republican president with less than 40% of the popular vote.