BUT!

While Bebop and Lain are definitely must-sees, I would wait till you're further on in your anime career to watch them. Bebop is too good; you shouldn't start out with something that impressive or other animes won't be interesting to you. And Lain is too messed up to be starting off with.

I recommend Bubblegum Crisis 2040, Magic Users Club (or something like that), and Kodomo no Omocha. Now, Child's Toy hasn't been released yet in the US, but you can get very good and inexpensive fansubs (VHS) at animemetropolis.com . Those guys are cool, very clean, they won't sell fansubs if the series has been released in the US; their sole purpose is to gain popularity for animes that have not yet reached the US market, so when they do, they'll be very popular.

We saw some of an anime called "Crest of Stars" at Otakon, and it looked very decent. Go for it. "Revolutionary Girl Utena" is deceptive, cause it's made by the people who did Sailor Moon, but it's not stupid at all. It seems like it might be for about two episodes, and then you're like... holy hell... what happened? Cause it turns really dark and serious really fast, and only gets more so as the series goes on.

Of course, Tenchi and Gundam series are very popular, but they're also very extensive, and you'll end up shelling out a lot for them.

Oh, "Master of Mosquiton" is also quite good.