A few of the above posters already given you several good shows, but that depends mostly on your proper tastes.
I will give some suggestions based on "certain" genres;
Action anime:
Watch either Tengen Toppa Gurren Lagann (which you can watch either the series, whose enlish-dubs seem better than the subtitled version, or blow it off quickly with the shortened, even-more-over-the-top subbed-only movie adaptation)
....and the Dragonball Z movies (the first three are the only ones done well, IMOP).
Off-beat anime:
Fooly Cooly or FLCL for short. It has quirky and with extremely eccentric loony toony battles. It also deals with the underlying aspect of coming-to-age.
Balanced anime (action, comedy, drama, etc):
Full Metal Alchemist, Trigun, Cowboy Bebop, and Samurai Champloo.
Thriller:
Death Note and Code Geass are pretty good at this, but if you want something more sophisticated and grounded-to-reality, then go see Naoki Urasawa's Monster.
For kids:
I'd say Digimon over Pokemon, since the former tends to be more bearable to watch with the young ones, and only the earlier seasons of both.
If you don't mind subs, then go see Heroman, which the first anime penned by Marvel creator Stan Lee, of Spider-Man and X-men fame. It follows anime tropes to T, which might turn some people off (it almost did for me). But actually became a bit endearing in its own way.
Psychological anime:
Neon Genesis Evangelion, starting with the original series, then its follow up movie, End of Evangelion. From there you can decide whether to watch the Rebuild versions, which some theorize might be sequel than a retelling of the series.
The Big O is actually more noir-styled than psychological, at least before the later half of the series. For the most part, it feels like Batman: The Animated Series but with giant fighting robots. In other words... AWESOME!
Long-running anime (100+ episodes):
Watch Dragonball starting with original series, with Goku as a kid. It is a vastly more endearing and heart-warming than what it becomes later, which is essentially a slugfest (which ain't bad, mind you). From there, watch the Dragonball Z Kai version, which cuts out most of the filler.
One Piece, with over 500+ episodes including 10 films it has one the bigger cult followings in long-running anime, and for good reason. I'd suggest skipping some of the filler content, though.
