In my opinion, Evangelion is not worth the time spent to watch it. It's regarded as a standard by many, but the characters are very shallow and the plot never seems to advance.
My entry into the world of anime was Ranma 1/2, which is campy and a bit sappy at times, but does contain a larger variety amongst the cast of characters. you'll find that the vast majority of anime series are fairly formulaic. This means that if you find a formula you really enjoy, there will always be something great for you to watch!
My favorites are [in no particular order]: Spirited Away, Time of EVE, the first 2 ep's of FLCL, the Read or Die movie, and Bamboo Blade. I really find that the first 1 - 3 episodes of a series are the best that series will have to offer; maybe that's because those are the beneficiary of all of the pitch work done to get the show sponsored and picked up by a network. After they are shown, the production houses rush to complete the projects ASAP.
There's a bunch of stuff on Netflix that rotates from month to month [watched all of Mezzo on there recently], and crunchyroll.com streams shows that aren't commercially available in the states [and therefore they are not breaking any copyright laws to do so]. Go there and watch Time of EVE now!