Some that haven’t been mentioned yet—correction, make that hadn’t been mentioned, when I first started writing this. Oh well…
If you don’t mind venturing into the YA section of the bookstore/library, I highly recommend Lloyd Alexander’s Chronicles of Prydian. (It’s a 5-book series, each of which is somewhat self-contained, but they tell a bigger story and should be read in order.) (Okay, on preview, I see this one has been mentioned, but it’s definitely worth mentioning again.)
Ursula K. LeGuin’s Earthsea Trilogy is rightly considered a classic. (She later went back and wrote more; by now there are six volumes, but you can stop after the first three, or even after one or two.) There are wizards and dragons and fun stuff like that, but it’s more philosophical and mystical than it is action and swordplay.
C. S. Lewis’s Chronicles of Narnia and J. K. Rowling’s Harry Potter books are, of course, enormously popular and might be close to what you’re looking for.
Meanwhile, back in the adult section…
Fritz Lieber’s stories about Fafhrd and the Grey Mouser are classic Swords-n-Sorcery; they’re one of the first things Terry Pratchett parodied in Discworld. The stories, of varying lengths, have been collected in six volumes that all start with the word Swords.
The first thing I read after finishing The Lord of the Rings for the first time was Stephen R. Donaldson’s Chronicles of Thomas Covenant. Some people love it, some hate it—partly it depends on how you react to Donaldson’s prose style and to having a protagonist who isn’t very likable. But I’ll give Donaldson this: he did write a series that is both very Tolkienesque and highly original. Donaldson also wrote Mordant’s Need, a separate fantasy in two large volumes (takes several hundred pages to really get going, but worth it if you can stick it out that long), and two impressive volumes of short stories.
I’m sure there are others that I’m not thinking of right now. In fact, there have been threads like this before, so if you don’t get enough good suggestions out of this one…