So basically, this is to me the most important question when it comes to history. It is precisely "How do we figure out what actually happened?" that drives history. I'm not a professional historian, but I'll share some advice about sources from my work in journalism (and plagiarize a few techniques):

1. The biggest reason you're going to see discrepancies in sources, especially old sources, is not going to be malice, but limited information. Take your example of Lexington: Any one person probably is not going to have a bird's eye view. Even the militia leaders on one side aren't going to know exactly how many men they have - and all bets are off for estimating how many men the other side has.

2. People also have foggy, terrible memories that (as you said) are susceptible to revision to fit what they want to believe. "Who shot first?" is a tricky one (I'm not too familiar with this Battle in particular) but you'd never get all the sources to line up. The best you can do with conflicting sources is A) Admit that the answer isn't clear and B) Evaluate them in totality.

So how do we get to evaluating sources, especially from the past when people aren't writing necessarily for us nosy historians? For me, I follow the general rules of Point of View (I don't quite remember where I read about this but I find it very useful):

Speaker: This one's pretty obvious, but also consider what bias the author is going to have. Are they British? American? Are they poor, rich, were they even there?

Purpose: What did the author hope to accomplish by writing what they did? Is it an obituary, a death tally, a journal entry, a letter, or a speech?

Occassion: What is the background of why the author wrote what they did? Are they a record keeper, were situations of this type usually recorded? Important to keep in mind here are the facts you do know, the Big History affecting any event: For the Battle of Lexington, yes it's the Revolution, but it's also 18th century warfare, the Enlightenment, and growing rapid changes in "America."

Audience: This is both the direct audience (who is the author writing to) and the indirect audience (who is the author writing for). Consider Paul Revere publishing his print of the Boston Massacre; Sure it's for a newspaper audience, but who is he trying to reach? Impassioned patriots, apathetic colonists, British loyalists?

Tone: How does the author write? This probably requires the most subjectivity but can be important nonetheless. Very important to consider here are the changes in style over time: An 18th century source might be tonally dissonant to a 21st century reader.

By synthesizing different sources together, you can get a pretty good idea of what happened. The last thing you want to do as a historian is give certainty to an event we don't have certainty for: Sometime's it's okay to see "We don't know all the facts, but maybe it went something like this..."

I hope this helps, and I encourage you to read as much as you can, and be lightly critical of it all!

If you're interested in Lexington, I have an answer about that here. The tl;dr is that battles are stressful and loud, a lot of men who were in the militia had been drinking off and on that night, and that people only see a part of the larger picture at any given point. It's a known fact in neuroscience that the higher levels of our brains shut down when we're under stress -- this is why if you've been in a car accident or another situation like that, you think time stood still; you lose access to part of your brain that records time passing as a stress response. Seeing British regulars coming at you, or seeing a crowd of militia blocking your path, can give you tunnel vision.

Anyhow. To quote the pertinent parts:

The local militia are forming, but there are relatively few of them on the ground -- Sylvanus Wood, a militiaman from Woburn, counted only 38 as he was falling into line, and most estimates say around 60 or 70 men stood their ground there that day. But from the British end of the field, the apparent numbers of the militia would have been increased by spectators milling around the Buckman tavern to the left of the militia formation and the horse stables and home to the south of the Concord road, to the right of the formation.

As Revere passed through the line, he heard Parker giving orders to his men to let the regular troops pass by, saying "Don’t molest them, without they being first." It was important to the Patriot leaders on this day, as it had been at Salem and other confrontations before, to not be seen as the first men to fire. Recollections of what was said were confused, but Parker chose to hold his ground there, even as some local men were arguing against it.

As the head of the British formation approached the Commons, they faced a fork in the road. The left turning would take them towards Concord, their eventual target that day, but would leave the militia on their right flank; the right turning would take them between the meeting house and the tavern, on the Buckman road, and head them directly toward the militia.

The Marine lieutenant leading the column, Jesse Adair, made a snap decision to turn right, and brought the three leading companies of Regulars along with him. Major John Pitcairn, in overall command of this detachment of troops, led his part of the line down the left fork and halted them there, then galloped across the Commons towards Adair's men.

The two leading companies of men with Adair, the light infantry companies of the 4th and 10th regiments of foot, follow him across the Commons and about halfway toward the militia. They deploy from column into line formation as they do so, one of the complicated military evolutions of the 18th century that British soldiers practiced to exhaustion. The British soldiers were cheering as they did so, adding noise to the already-confusing battlefield

Pitcairn, possibly with some other officers, rode up to the militia, pistol in hand, and ordered them to disperse. His words were recorded differently by several men there that day -- unsurprising, given the stress of the moment and the subsequent events. Some remembered him saying "Lay down your arms, you damned rebels!," while others remember something like "Throw down your arms, ye villains, ye rebels" or "Ye villains, ye rebels, disperse, damn you, disperse!"

Upon hearing those words, Parker immediately ordered his militia to disperse.

In the time-honored tradition of military affairs, not all of them got the message. Some men stood their ground.

At this point, a shot was fired.

We have several pieces of evidence to say where the shot came from, but they disagree on the particulars.

Revere and Lowell have crossed the road to the west of the common and are moving between a home and a blacksmith shop. Revere later testified that as they did so, he heard what he thought was a pistol shot. The line of militia was blocked from his view, so he couldn't see where it came from.

On the British side, Lieutenant Edward Gould was near the center of the British line on the Common. He also heard a gunshot as the two lines drew near, but he did not see where it came from, and had trouble hearing it over the shouts and cheers of the British troops.

After the battle, several British officers said they saw a "provincial" fire at them from behind a stone wall, while some of the regular troops said a shot came from the corner of the Buckman tavern.

The Lexington militia thought that one of the mounted officers fired at them -- either Pitcairn or one of his men -- though Pitcairn testified after the battle, and was supported by his officers, that he had ordered his men not to fire.

One thing that seems to have been generally agreed on is that neither the regular soldiers, nor the militia, fired first -- the shot came from the spectators on the outside of the Common, or from an officer's pistol.

The likeliest explanation is that several shots were fired in quick succession, after an initial shot was fired. Men who could only see a portion of the field would have reported what they saw. As far as why someone fired an initial shot, we should recall that men on the Patriot side had been drinking heavily off and on all night in the Buckman tavern, while several British officers were mounted and holding pistols which could go off without much provocation (as could many 18th century firearms).

There's no doubt that after the initial shot or shots, the British regulars began to fire without orders, first singly then by platoon as they were trained to do. They did not fire aimed shots, but reloaded, presented arms and fired as quickly as possible. As they did so, some of the remaining militia were hit and wounded or killed. Some fired back, while others scattered, as did the spectators around the Common. Revere and Lowell, having moved farther west, were in the line of fire, and Revere testified that musket balls flew around them as they struggled with the trunk.

(Much more at the link ...)