I don't know how well respected it is, but Vicki L. Eaklor’s Queer America: A People’s GLBT History of the United States was used as the main book for an LGBT history undergrad course I took in 2014. It seems like a good place to start.

This post is totally fine!

George Chauncey's Gay New York is fairly foundational as a study. Keep in mind a few caveats: first, it focuses on New York, which was definitely the United States' gay metropolis for most of the time of his study, but will not be representative of other areas in the US (especially rural areas). Second, Gay New York is very much a history of gay identity (that is, homosexual male identity), not lesbian or other LGBTQ identities.

Some books off the top of my head include; Martin Duberman's Stonewall about how & why the Stonewall Riots happened, Randy Shilts' The Mayor of Castro Street: The Life and Times of Harvey Milk about the first openly gay US politician.

Another book by Shilts that you might enjoy is And The Band Played On: Politics, People & The AIDS Epidemic Elizabeth Lapovsky Kennedy and Madeline Davis's Boots of Leather, Slippers of Gold is a good book about Lesbian communities pre-stonewall.

This is a little off-topic but you might like Anthony Bogaert's Understanding Asexuality if you're interested in learning more about gender identities. You can also read my answer on Asexuality here It's a bit psychological but you might like it. I'd also suggest contacting /u/Cephalopodie as she specialises in LGBT history :) But if you're interested in the psychological side of Gay rights feel free to ask me :)

I second Gay New York and most of the stuff /u/subs-man mentioned (haven't read Mayor of Castro Street or Understanding Asexuality so can't vouch for those.) I would offer a qualified recommendation of And the Band Played On - it's a really important work, but I wouldn't recommend it as your introduction to the AIDS crisis. It was written as a work of investigative journalism in the mid 1980's, and as such just doesn't stand up to historical accuracy.

I have a lot of books I can recommend relating to LGBTQ history, but it all kind of depends what sort of narrative you're looking for. Where you draw the line between what is a part of "the movement" and what is just "history" is an exceedingly complex and often pretty fraught issue. I'm going to throw some more general and/or activism-specific titles in your direction, but feel free to ask for more specific areas if there's something you want to know about.

George Chauncey's Why Marriage contextualizes the same-sex marriage movement.
Lillian Faderman's Odd Girls and Twilight Lovers is a good overview of American lesbian history
Allan Berube's Coming Out Under Fire explores gay and lesbian experiences during WWII
Michael Bronski's A Queer History of the United States
Deborah Gould's Moving Politics is a really exciting and intellectually complex look at ACT UP
Susan Stryker's Transgender History re-centralizes the experiences and vital importance of trans folks in and around the movement.

IF you like documentaries, there are a few good ones:
The Life and Times of Harvey Milk
United in Anger
How to Survive a Plague (recommended with reservations: the narrow scope leads to some significant sins of omission)
Larry Kramer in Love and Anger
Vito