According to the Word of God, the Austin Powers series is a parody to all of the British James Bond-knockoffs in the United Kingdom in the late '60s and early '70s that Mike Myers used to enjoy, though there are plenty of James Bond references as well.
The opening of the movie, with Austin running in the street, followed by hundreds of screaming fans, is a direct reference to The Beatles.
Austin's cover identity, that of a ridiculous fashion photographer, is taken from the photographer character from prototypical "swinging '60s London" film Blow Up. His over-the-top "and now I'm spent" photography style is a direct lift from that film.
An obscure audio shout-out: Austin's ringtone is the same as Flint's ringtone in the 1966 superspy parody Our Man Flint.
Alotta Fagina and Random Task's names are both parodies of character names from Goldfinger: Pussy Galore and Odd Job, respectively.
During the final battle, the song Secret Agent Man plays. This was originally the theme song to Secret Agent.
There's the shout-out that Austin made to the 1960's James Bond knockoff In Like Flint. While he's watching it, Austin even says "That's my favorite movie!" Years later, a magazine print ad for a DVD re-release of the Flint series gave a shout-out in turn by using that line as a pull quote.
Star Wars is frequently referenced in the second film as well.
The trailer deliberately misled moviegoers into thinking that it was a preview for the then-upcoming The Phantom Menace. Instead of Palpatine manning the Death Star, though, it's just a chair reveal of Dr. Evil. The following tagline echoes that of The Naked Gun:
"If you see only one movie this summer, see—! ...Star Wars.
"The moon unit will be divided into two divisions: Moon Unit Alpha and Moon Unit Zappa."
"Anyways, the key to this plan is the giant laser. It was invented by the noted Cambridge physicist, Doctor Parsons. Therefore we shall call it... The Alan Parsons Project!"
The music during the intro when Big Boy floats past an astronaut is a parody of the soundtrack to You Only Live Twice.
When Dr. Evil is spinning out of control on his chair, he declares that he needs an old priest and a young priest. This is a reference to The Exorcist, where an old and young priest attempt to stop a possessed girl's head from spinning around.
Austin Powers in Goldmember
It has an homage to Silver AgeSuperman and Lex Luthor going on, with Young Austin attending a spy "academy" with Number Two and an adolescent (and already-bald) Dr. Evil. When Austin wins the coveted International Man of Mystery award, Dr. Evil vows revenge.