Here are the "modern" books that I picked as a sampler of science fiction and fantasy. I thought of it as one of those cheese plates or chocolate boxes. It has a little of everything, and I tried to hit all the big notes, even those that aren't to my taste (some people like coconut or nuts, and even though Neil Gaiman isn't to my taste, he is still one of the big tastes in fantasy). I wasn't trying to be exhaustive (this has a startling lack of Patricia Wrede, for instance). I thought I'd post this here, in the hopes that someone might find or book, or that you, dear reader, could point out any glaring oversights.
Since this was a sampler, I tried to pick each of a sort of book.
I, Robot - Issaac Asimov - This is a clever set of stories that depend on the three laws that Asimov created to define the behavior of robots. Much of modern science fiction builds or interacts with Asimov's conception of robots.
Fahrenheit 451 - Ray Bradbury - It's hard to know where to start with Bradbury. I think his work is pretty consistent, though, so if you like one, you're likely to like most. I've suggested Fahrenheit 451 because it is one of the best known, and because I enjoyed it the last time I read it.
A Wizard of Earthsea - Ursula LeGuin - LeGuin is one of the big names in science fiction, and quite deservedly. This is about names and how they interact with the thing that is named, and is set on various islands.
A Wrinkle in Time - Madeleine L'Engle - Many of my friends like this book, and although it has never quite clicked for me, I still think it worth reading, for Mrs. Whatsit, Mrs. Who, and Mrs. Which.
The Abhorsen Trilogy (beginning with Sabriel) - Garth Nix - I reread these quite often. Garth Nix does everything well: characters, description, plot, action, and worldbuilding. Although this trilogy isn't about vampires, it pairs well with Dracula. Both are about the fight of light and law against dark and death.
Alphabet of Thorn - Patricia McKillip - I had a hard time choosing which book of McKillip to suggest, because I like so much of her work. She writes lovely fantasy that draws heavily from fairy tale imagery.
Coraline - Neil Gaiman - I'm not sure the best place to start with Gaiman. So many of my friends adore him that I thought I should include Coraline. Quirky and a bit dark. It is also a wonderful stop-motion film. (Or, you might prefer to try the BBC radio play of Neverwhere, which I have and is wonderful.)
Trading in Danger (the first book in the Vatta's War series) - Elizabeth Moon - If you want to try space opera, start here. Action, adventure, space pirates, and saving the galaxy.
The Human Division - John Scalzi - more excellent space opera, but this time centered around diplomats and told in a series of linked short stories.
Howl’s Moving Castle - Diana Wynne Jones - A moving castle, an eldest daughter who sets out to seek her fortune, transformations, and a clever use of a John Donne poem.
Going Postal or Wyrd Sisters - Terry Pratchett - Quirky, hilarious fantasy that pokes fun at fantasy and has hilarious footnotes. Also a bit of a social commentary.
Tam Lin - Pamela Dean - a novel length re-telling of Tam Lin in a modern-ish college setting.
Beauty - Robin McKinley - a novel length re-telling of Beauty and the Beast in a fantasy setting.
Restoration of Faith (short story available here) - Jim Butcher - Joel knows about Dresden, and as I've yet to read the series, I included his suggestion. Try Dresden if you want occasionally humorous modern fantasy centered around a private detective. From what I've heard, the series improves quite a bit after the first couple of books.
The Last Unicorn - Peter S. Beagle - A fairy tale, of sorts, but this story centers around a unicorn. A haunting, beautiful, slightly melancholy story. (You can also watch the lovely animated movie.)
Jonathan Strange and Mr. Norrell - Suzanah Clarke - Victorian England, the Napoleonic Wars, the return of magic, and the Raven King. Not to mention wonderful footnotes that fly off on tangents to tell stories of their own.
the Harry Potter series - J.K. Rowling - The best is Harry Potter and the Order of the Phoenix, but it is the fifth book and you really have to start at the beginning and read the entire series. Rowling writes characters very well, and is concerned with the disenfranchised.