The following recommendations are for both the manga and anime versions since one is based on the other and they're both excellent.
Rurouni Kenshin:
Meiji Kenkaku Romantan
This is an often overlooked series in North America despite it being hugely popular in Asia. It's the story of a
Hitokiri's (samurai/assassin type person) life, ten years after the
Bakamatsu (late 19'th century civil war that resulted in a revolution), of which he played a pivotal role. It's an action/drama about redemption and forgiveness. It's entertaining at first and starts getting good at episode 8, then gets
really good at episode 28. It is my personal favourite so please give it a try...
One Piece
This is currently (or was, I might be a little out-of-date) the most popular manga and anime in Japan, so you might want to give it a try. In this world, 50 years ago, the great pirate
Gold Roger was finally captured by the marines. At his execution, he announced that his greatest treasure, simply called
One Piece, was hidden somewhere along the
Grand Line, the most dangerous sea in the four oceans. Shortly thereafter, everyone and their dog put together a ship to go looking for One Piece and a Great Age of Pirates has begun!
One Piece is the action/adventure story of a young man and his pure determination to gather a crew, find One Piece and become the Pirate King...
Naruto
This is quickly becoming popular in both Japan and North America. The
Nine Tails Demon-Fox was so powerful that a swing of its tail could cause mountains to crumble and tsunamis to flow. Luckily, one ninja in the village of
Konoha, the fourth
Hokage (leader of the village), sacrificed his life and managed to seal the Demon-Fox into the orphan child,
Uzumaki Naruto, the title character. This is the story of Naruto growing up as a ninja among other ninja of varying skill. It's an action story with some very good suspense, surprises, and character insights. I personally think the manga is much better than the anime but the anime is quickly becoming popular so perhaps it's just me...
These are my biggest recommendations.
There are many other series that you may or may not like, depending on what you're looking for. A lot of people will recommend
Cowboy Bebop but I'm not a big fan. It's very stylish but the characters are rather flat and the stories are unexceptional. For something just as stylish (but in a totally different way) with a better story would be the action/adventure anime
Read or Die OAV (not the TV series, which drags a bit).
Trigun is a good action/drama pseudo-western. For some reason, the anime is much more exciting than the manga.
Azumanga Daioh is a
kawaii (cute) and funny story about a bunch of girls, diverse in personality, going through highschool together. The manga is supposed to be like a newspaper serial (like
Family Circus) type of thing (I've never read it) but the anime flows like a story.
Full Metal Panic: Fumoffu is a hilarious (and mildly farcical) comedy about a militarized kid trying to go through an ordinary highschool. Please don't confuse this series with the original
Full Metal Panic, which is a serious action series.
Again, I have more recommandations I could make but I've limited myself to somewhat recent series you're likely to find at your video store or online.
Good luck!