As we answer this question, we must keep in mind that the American rebels were not one monolithic group that all had the same goals. This means that different rebels would have been making different points. Let us look at the two main points that were being made.
One point that was being made was that people ought to be able to rule themselves. The colonists did not like being ruled by people in faraway England or by those appointed by the people in England. They wanted local elites to be treated the same as English elites. That is, they wanted their own elites, rather than elites from England, to govern.
A second point that was being made is that democracy is the proper form of government and that monarchy is a bad system. Some of the colonists were rebelling largely because they felt that all the people should be allowed to govern themselves. They were not trying to ensure that American elites governed. Instead, they wanted Americans of all sorts to be involved in self-government.
Thus, there were at least two points being made. One was a point about self-government while the other was a point about true democracy.
One correction I want to make real quickly is the date for the American Revolution. If we define it as when the Revolutionary War took place, it began in 1775, not 1774, with the Battles of Lexington and Concord.
I think the main point the Americans who rebelled agains Britain were trying to get across was that everyone has certain rights that need to be acknowledged. This is evident when taking a look at the Declaration of Independence. The very first sentence in the preamble is the forever famous line, "We hold these truths to be self-evident, that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." The document makes it known right away what these rebels care about, and later on states that it is the government's duty to ensure these rights. The list of grievances against the King indicate that independence comes from government's failure to protect these rights, thus showing how much rights are valued.
Along those lines, the rebels also tried to make an equally important point about what is the correct form of government. Obviously having a monarchy just wouldnt do. Rather, the particpants of the Revolution wanted a Republic, where officials are elected to represent the people. This is stressed multiple times through the revolutionary period. The phrase "No taxation without representation" was common in the period leading to the revolution, the Articles of Confederation was highlighted by a weak federal government, and later, the Constituion would split government into the three branches.
Overall, I think you could maybe generalize the main purpose of the American Revolution as one where rebels wanted to instill a government represented by the people, in order to protect people's rights.