If you want a perfect intro to Anime I strongly suggest checking out Cowboy Bebop. It's only 26, 1/2 hour episodes. It covers the range of Anime from silly, funny, amazing action, characters that you really come to care about and some really heart wrenching stuff too. It's a "best of the best" series.
Another reason to watch Bebop for a starter is that the english dubbing is done VERY well. Sometimes they change entire dialog sections, and modify the meaning of the story when they dub to english. There are series that suck when viewed dubed but are great sub-titled. Often english voice actors don't "match" the characters and that is distracting as hell.
Most Americans exposure to Anime is limited to stuff like Speed Racer and Gundam. There are much better series out there.
Another favorite of mine is Tenchi Muyo! (the first series, not the second one). Tenchi is one is better viewed subbed rather than dubbed.
For most people it's either an "I LOVE THIS STUFF" or a "What the hell is this shit?" type of reaction. One thing to keep in mind is that you can't begin to get a sense of a series with just a few minutes. You HAVE to watch it for a bit and then the "ahhhhhh!" reaction will come.
Oh, another great one to start out with is the film "Ghost in the Shell".