If you like Star Trek: David Mack's Destiny trilogy, starting with Gods of Night. This is easily the best post-movies series available. Or if you want a novel-only series, try the Vanguard series, starting with Harbinger (also by David Mack - he writes some epic stuff). Vanguard is in the Original Series time frame.
If you like Star Wars: Heir to the Empire by Timothy Zahn or Michael Stackpole's X-Wing books. (Fair warning - a lot of post-Episode VI novels feature the same characters over and over again - Luke, Leia, Han, etc. This is my complaint about Star Wars - slow to add other characters. But I've still read lots of them.)
Totally dated but still good: Isaac Asmiov's Foundation Trilogy (spans centuries, makes you think).
Absolutely crazy: anything by Neal Asher. Gridlinked is the first in his Agent Cormac series. These are complex but totally worth it.
Speaking of complex, Pushing Ice by Alastair Reynolds is also outstanding. It has a lot of emotion to it. I'm currently reading House of Suns by Reynolds and it is good (as of page 250).
If you like military stuff, try Starship Troopers by Robert Heinlein. He wrote a lot of other stuff that I haven't got around to yet.
There are a lot of Philip K. Dick short stories you can get for free. He's another one of the classic sci-fi authors. I started with The Skull.
Ender's Game by Orson Scott Card is very good. The sequel, Speaker for the Dead, is also great. The third book, Xenocide, is terrible.