I was lucky enough to study Science Fiction at University, so I had to read tons of the stuff. Here are a few novels that you might enjoy:

Frankenstein - Mary Shelley
The Time Machine - H.G Wells
A Door to Summer - Robert Heinlein
I Robot - Isaac Assimov
The Hitchhiker's Guide to the Galaxy - by Douglas Adams
Flowers for Algernon - by Daniel Keyes
Do Androids Dream of Electric Sheep - Phil K his last name I can't write on blurtit, but it means male member and is short for Richard.

These are some of my favourites, I'm particularly fond of Heinlein's book, I've bought it a dozen times because I keep giving it away.

Frankenstein is very early Science Fiction, and Keyes Flowers for Algernon, really does pre-date and pre-empt much of modern science's attempt at cloning and transplants etc.

Some books I've heard are good but I've never read:

Dune by Frank Herbert
Starship Troopers by Robert Heinlein
Neuromancer by WIlliam Gibson.

 To epictetus's list, I would add:

THE MARTIAN CHRONICLES by Ray Bradbury, which has a a poetry you don't always find in science fiction

THE LEFT HAND OF DARKNESS by Ursula Le Guin, which is an intriguing exploration of sexuality in a way that only science ficiton could do

NEUROMANCER by WIlliam Gibson, which is the work that gave us the word "cyberspace" and rocks and rolls

THE SPACE MERCHANTS by Frederick Pohl and C.M. Kornbluth, which is a sly satire on advertising

A CANTICLE FOR LEIBOWITZ by Walter M. Miller Jr. about the world after the bombs drop

THE GIVER by Lois Lowry is technically a "young adults" novel, but it's must reading for anyone with even a casual interest in science fiction. It's about a society in which EVERY aspect of life is controlled and predetermined, and what happens when something happens to one young boy that causes him to question that society.

1984 by George Orwell is the classic dystopian future. It's depressing but important.

A CLOCKWORK ORANGE by Anthony Burgess is another downer that's brilliant (especially in its use of language).

Hope this helps!