The first to get much notice in the U.S. was Frank Kameny, who was dismissed from his job as an astronomer from the US Army Map Service in 1957 for being gay.

That led him to a life of activism, including filing to argue his case against the US Civil Service before the Supreme Court in 1961. It was the first claim of homosexuality as a civil rights issue, even though he was denied a review. Kameny also led one of the first public demonstrations by gays and lesbians in DC in 1965.

Earlier still was the communist Harry Hay, who co-founded the Mattachine Society in November of 1950 in L.A. Its history makes for an interesting read as well, as its development was closely tied to the Red Scare. Eventually the 50s ended and the Mattachine Society became more of a group purely about LGBT rights. Kameny, in fact, led the DC chapter.

EDIT: Also see comments about Emma Goldman below, who can probably lay claim to being the very first.