Fantasy
I really like China Miéville's Perdido Street Station but the actual plot doesn't start until halfway through the book. The first half goes into a lot of detail about the characters and the city of New Crobuzon, and things happen, but some readers get overwhelmed. If you like detailed world-building you'll probably like it though, but maybe read the first Dune book first and see if that kind of thing is your speed.
I cannot recommend Havemercy by Jaida Jones and Danielle Bennett enough. It uses the shifting character perspective like Game of Thrones, but from first-person, and it has a rather unique premise and setting (fantasy czarist Russia using magic clockwork dragons in a war against fantasy China).
John Marco's Tyrants and Kings trilogy (starting with The Jackal of Nar) was also quite good.
I love Steven Brust's Vlad Taltos books a lot, especially the earlier ones. Most of the books stand alone, so you should be able to pick up any of one of the first couple and not feel lost. And they're short.
I think Robert E. Howard's work is mostly in the public domain in much of the world--I picked up a three-volume collection of the Conan stories just last year. The Howard Conan stuff is all good--Conan works best in the short story format. I wouldn't bother with most of the novels written by later authors--it's formulaic and predictable. Other Howard stuff isn't bad either. I like the Kull stories, myself. Kull is like Conan but dumber, who mainly succeeds at the task of ruling the greatest kingdom in the land because of his two extremely clever friends.
Fritz Leiber's Lankhmar books are available through Amazon for a good price, and I highly recommend those too. If not, beg your local library to stock them.
Neil Gaiman is, of course, the king of dark modern fairytale/urban fantasy.
Science Fiction
SF is a bit harder to recommend since there so many different subgenres, as well as the hard/soft divide.
Some essentials for me would include:
You've already mentioned Dune. Read it. I consider the first trilogy some of my favourite books of all time.
Someone already mentioned Heinlein. Recommend you start with one of his earlier, standalone books: one of the so-called "juvenile" books, or Starship Troopers, The Moon is a Harsh Mistress, or The Puppet Masters. Maybe Stranger in a Strange Land. Some of the later books like The Cat Who Walked Through Walls and Number of the Beast make too many references to his "multiverse"
Neuromancer by William Gibson, the book that basically kicked off the cyberpunk genre.
Snow Crash by Neal Stephenson, the book that basically capped off the cyberpunk genre. Plus it's awesome.
Alastair Reynolds writes hard SF space opera, and he really works at creating a plausible universe where it takes centuries to travel between stars, and where space battles are waged at ranges of 2 AU, but it all seems to function.
Robert J. Sawyer writes hard SF but makes it mostly accessible, and tends to ground it in the here and now, or close to it. Calculating God, Rollback and Flash Forward (which is very different from the TV series) are good standalone books to try.
So far, I've loved anything by John Scalzi. He creates likeable characters (if he has a fault, it's that some of them are too likeable), writes great humourous dialogue, and still does a decent job at plotting and world-building. Agent to the Stars or Old Man's War are good places to start.
A couple of Ursula LeGuin's classics would include The Lathe of Heaven and The Left Hand of Darkness.
The second one is written by Isaac Asimov and Robert Silverburg called Nightfall(was based on a short story by Isaac).
The last one is called The Lovers by Philip Jose Farmer, but it has a fair bit more to it than that. Here's a link [http://www.amazon.com/Lovers-Philip-Jose-Farmer/dp/034528691X/ref=sr_1_1?s=books&ie=UTF8&qid=1281464607&sr=1-1] since it's a bit harder to find.
