Ghost in the Shell
(the movies and the series) is probably the go-to for serious science fiction anime. It's about an elite team of cyborg-ized special counter-terrorist agents.
is the go-to for just plain fun
science fiction anime. It has a little bit of everything - space bounty hunters, buddy-cop, Star Warsy space battles, con games, spoofs, film noir... Lots of good humor and a GREAT jazz score by Yoko Kanno (as does GitS). Bebop is the anime that got a lot of Americans hooked on anime.
has been compared to 2001 in its near-future-space styling and story. A very good story about average folks who just happen to work in orbit, collecting space junk, and their hopes and dreams.
The Last Exile
is a killer steampunk aerial adventure series with vast fleets of antigrav battleships and fighters that look like 1936 Chords that fly.
is a cartoony spoof of the 1960s spy genre, but set slightly in the future, with more cyborg spy babes.
, if you like giant robo action and being confused beyond hope.
And of course, almost any film by Miyazaki:
Nausicaa of the Valley of the Wind
My Neighbor Totoro
Kiki's Delivery Service
Castle in the Sky