As far as major political parties, the Democrats have been around longer than the Republicans. However, like you mentioned that the Republicans have changed significantly over the years, so have the Democrats.

The Republican Party which exists today was only founded in 1854, shortly before the Civil War, as an anti-slavery party. It shouldn't be confused with what historians now refer to as the Democratic-Republican Party (which was contemporarily known as the Republican Party) founded in 1791-92 by Thomas Jefferson and James Madison as an anti-Federalist party. The Democratic-Republicans adopted the 'Democratic Party' title in 1844, and continues to this day.

The other major parties of the early USA have all long since disappeared. For example: the Federalist Party had been dissolved by the Jacksonian era; the Whig Party was influential in the first half of the 19th century but was dissolved before the start of the Civil War.

Also worth noting, just as interesting context, that several of the Founding Fathers were decidedly against political parties, and that the US constitutional system was originally designed without making allowance for them, and in an attempt to drive consensus-based government. The most clear example of this is the fact that, until the ratification of the 12th Amendment in 1804, the Vice-Presidency was held by the runner-up in the presidential election.

James Madison, 1787:

Complaints are everywhere heard from our most considerate and virtuous citizens, equally the friends of public and private faith, and of public and personal liberty, that our governments are too unstable, that the public good is disregarded in the conflicts of rival parties, and that measures are too often decided, not according to the rules of justice and the rights of the minor party, but by the superior force of an interested and overbearing majority.

Source: James Madison, Federalist No. 10 (1787)

John Adams, 1780:

There is nothing which I dread so much as a division of the republic into two great parties, each arranged under its leader, and concerting measures in opposition to each other.

Source: Susan Dunn, Jefferson's Second Revolution (2004), 39.

George Washington, 1796:

The alternate domination of one faction over another, sharpened by the spirit of revenge, natural to party dissension, which in different ages and countries has perpetrated the most horrid enormities, is itself a frightful despotism. But this leads at length to a more formal and permanent despotism. The disorders and miseries which result gradually incline the minds of men to seek security and repose in the absolute power of an individual; and sooner or later the chief of some prevailing faction, more able or more fortunate than his competitors, turns this disposition to the purposes of his own elevation, on the ruins of public liberty.

Source: George Washington, Farewell Address (1796)

Edit: typo.