I'm a big fan of Naruto and Bleach. Naruto in particular is probably my favorite anime of all-time.
Watch through the end of the Sasuke retrieval story arc which ends around episode 135, and then skip to Naruto Shippuden (a new series) where the Naruto characters have grown up a bit. The original series episodes go from 136 to 220, but they are just filler episodes and don't really move the main plot forward.
The English dub is, in my opinion, not very good. Watch it in Japanese with subtitles. Luckily I speak Japanese and my wife is Japanese so I can enjoy it without watching it in English. It is a fantastic series with a very positive message about how never giving up, and supporting friends and loved ones can make you stronger than you ever thought possible. The characters grow and mature and develop and there are awesome Dragonball esque fight scenes. One of my guilty pleasures when it comes to anime.
Bleach is similar in a lot of ways to Naruto. I also enjoy Eureka 7. Good character development and my wife really liked the romance angle with Eureka and Renton. My wife even dressed up as Talho at Sakura-con in Seattle. How cool is that!
Also Ghost in the Shell Stand Alone Complex is, in my opinion, simply the best realization of a cyber punk world ever done in any medium, print or film. PERIOD. Both series are good, but I like the second one a little more. I hate the movie though.
My wife is also a huge fan of Blood+. Its not bad, but I'm a sucker for over the top Dragonball style kickass action and its a little low key for my taste.
And finally, I have to recommend Avatar: The Last Airbender. Its not technically anime since it is produced in the U.S. (though animated in Korea), but it is a fantastic series that the whole family can watch. I love it and highly recommend to everyone. The first two seasons are done and the final season will premiere at the end of September.