Any and all "Best of" recommendations are inherently subjective and incomplete. This will be no exception. I've divided this short list into three sections: One for Sturgeon, one for Classic SF and one for personal favorites.
Section I: Theodore Sturgeon
Theodore Sturgeon was quite prolific, wrote under several pseudonyms and did much work on for tv which is not reflected here. Individual novellas and short stories are in collections and I'll note the anthologies for them, though much of the other stories are good as well. So let's start:
More Than Human, where individuals with odd powers band together to be more powerful as one entity. (This same concept would band mutants together as the X-Men.)
Slow Sculpture: Volume XII: The Complete Stories of Theodore Sturgeon, the 12th in a series collecting all his short stories, contains my favorite of his works: "The Widget, the Wadget, and Boff". And other good stuff too.
Microcosmic God: Volume II: The Complete Stories of Theodore Sturgeon contains, among other great stories, "Microcosmic God". Sixty years later it was parodied on "The Simpsons". Between "Microcosmic God" and Monty Python's "Life of Brian", you'll have what you need to know about how religions get started.
Section II: Classic SF
The Hobbit/Lord of the Rings by JRR Tolkien. I'm not going to separate the four books. I'm an sf reader, for the most part, and not fantasy, but H/LoTR remains on top. One of the few books I've read more than a couple of times. Note: The LoTR movies are fair adaptations, The Hobbit movies take much liberty.
The Science Fiction Hall of Fame, Vol. 1: 1929-1964 ed. by Robert Silverberg. Published in 1965, this remains an exceptional introduction to the field of science fiction.
The Science Fiction Hall of Fame, Volume Two A: The Greatest Science Fiction Novellas of All Time ed. by Ben Bova
The Science Fiction Hall of Fame, Volume Two B: The Greatest Science Fiction Novellas of All Time
The Moon Is a Harsh Mistress by Robert Heinlein. My favorite hard science fiction book, and remains on top of most lists. After this, if you want more Heinlein, go to his collection set in the same timeline, The Past Through Tomorrow.
The Best of Cordwainer Smith (Kindle edition). Where Heinlein's future history is based on Western Civilization, Smith riffs off Chinese motifs. One of the best stylists sf ever produced, his stories are a pleasure to read.
The Annotated Alice: Alice's Adventures in Wonderland & Through the Looking Glass by Lewis Carroll, annotations by Martin Gardner. The two Lewis Carroll stories are still read after 150 years while what they parody has been forgotten. Martin Gardner has done an excellent job digging up the deservedly-forgotten works. People reading at the time would have understood the pop culture references. But "Alice" works on so many levels. Here's how one of the more successful works of literature was constructed. Oh, and it's a lot of fun to read.
I'm going to stop here, and boy is it painful. There's so many great works that fall in the realm of fantasy and science fiction. These are a start, but only a start.
Section III: Personal favorites
Davy by Edgar Pangborn. A post-apocalyptic story set in the Catskills, where I grew up. Still hauntingly beautiful.
True Names: And the Opening of the Cyberspace Frontier by Vernor Vinge. I haven't read this edition, so can't speak to any story besides "True Names", but the 1981 novella remains one of the best computer stories around, and a seminal influence on cyberpunk and net policy.
Nova/Babel-17/Tales of Neveryon by Samuel R. Delany. I'm cheating here by naming three books, but hey, it's my list and Delany is my favorite living author. "Nova" is a far-reaching space epic, "Babel-17" is about how language influences thought, and "Tales of Neveryon" is possibly the best science fiction novel ever written even though it's in a fantasy setting. Dive into Delany.
Orphan of Creation, by Roger McBride Allen (Kindle Edition). Similar themes to "The Color Purple" but viewed from science fiction based on paleoanthropology.
Briar Rose by Jane Yolen. Adapting the "Sleeping Beauty" fairy tale to the Holocaust.
Way of the Spider by W. Michael Gear (Kindle Edition). The first of three in the trilogy. Military fiction is a major subbranch of sf and not my favorite, but these books are more than the battles and sweep of armies.
Stories of Your Life and Others by Ted Chaing (Kindle Edition). I'll stop with a more recent set of award-winning stories. I interviewed him at Minicon; an interesting guy (who's a tech writer first and does sf as a sidelight) who writes interesting stories.
I'm posting this on a LiveJournal blog so others can make comments. Who is YOUR favorite author, and which books by them would you recommend?