There are some series that are better to get into than others if you want to get into anime.
From my experience, I would probably stay away from Neon Genesis Evangelion, RahXephon, Serial Experiements Lain, and FLCL. This isn’t to say that these are bad series—in fact they’re some of the best—but they’re very odd, and could give you a bad impression of anime as a whole.
I’d recommend watching the following:
Martian Successor Nadesico: Good story, makes fun of anime itself.
Jin-Roh the Wolf Brigade: Alternate reality after WWII. More of a serious-style anime. It’s so realistic that you could see it made as a live-action movie and come off very well.
Hellsing: (warning: ultra-violent, not for the squemish)Takes some great liberties with vampires, but comes off with a highly enjoyable series. While some anime is okay for kids, this isn’t by any stretch.