Well, for a start, there are animes and mangas of all genres, so finding one you like is fairly easy, and there are a lot of animes
based on manga. All you have to do is pick a genre.
My own personal favourites are Rurouni Kenshin, which is now both a manga and an anime. It tells the tale of a reformed assassin in post feudal Japan, who has sworn never to kill again despite the troubles that keep haunting him from his past as he tries to live peacefully with new found friends. It actually has a bit of everything - action, comedy, romance, and samurai drama, so it has a lot of fans already.
Another one I would definitely recommend is Cowboy Bebop. This features a group of misfit bounty hunters as they travel through the solar system looking for the next job - and fighting over the cash. It's got a lot of great elements to it - comedy, action, sci-fi drama - and the music is excellent. The manga doesn't follow the anime, but rather provides a collection of standalone stories that don't require any knowledge of the series. Definitely one of my favourites.
What I've read/seen probably hasn't even scratched the surface of what's out there, but here are a couple of my suggestions:
Trigun - sci-fi action comedy-drama featuring a good natured gunslinger looking for something from his past.
Escaflowne - a lavish fantasy series, with a girl from modern Japan who finds herself in an unbelievable world, where her hobby - fortune telling - is actually real there,
Ghost in the Shell - a sci-fi crime drama, featuring a group of both human and cyborg detectives tracking down all sorts of crimes.
There are lots to choose from, and I'm sure that a lot of people will post their own preferences and recommendations here. Hope I've been of some help!