First two political parties of the US?

The first two US political parties were technically the "Federalists" and the "Anti-Federalists", who were later known as the Democratic and Republican parties.

federalist and anti-federalists
The first two political parties were the Federalist headed by Alexander Hamilton and John Adams and the NON-Federalists, who in time turned out to be the Democratic-Republic headed by James Madison and Thomas Jefferson. The Federalist will be absorbed by the Democratic Republicans after the War of 1812 and the Democratic Republicans will eventually split into the Democrats and the Whigs. The Whigs will eventually break up into several different parties that will eventually join together to found the Republican party in 1850.