I suppose after 10 years of watching the stuff I find it difficult to name any one (or even any 5) shows that one should watch. I've also fallen probably just beyond the "elitist" boundary as I watch everything in Japanese without subtitles until I have it on DVD.

In any case, as clasificado said there's a number of genres inside anime, as it is more a medium than a single genre itself. You'll have to peruse some suggestions and find out what you like.

If you're a pseudo-realistic Mecha fan, you'll have to check out the Gundam series, which spans some 30 years and 10+ series. A must-see is the original 1979 series, which while old animation wise is still quite good. Same goes for the original Macross.

If you saw and liked Cowboy Bebop (hard to believe the series is 12 years old now,) then you might want to look up the works of Studio Bones, which include shows like Rahxephon and Darker than Black. Most of the staff that worked on Bebop moved on to form Bones, and have worked on most of the shows it has produced since it formed, to varying degrees of success.

Gainax (Evangelion) seems to make the occasional comeback, especially with their recent series Tengen Toppa Gurren Lagann, which touches all the super-robot themes.

And you can't go wrong with the Ghost in the Shell TV series, both of which are far better than the movies.

There's an excess of material out there, and 3+ decades worth of stuff strewn with stuff that rivals the best western shows and some utter crap (a lot, as of late.)