As essential sets, I highly reccomend Initial D, Outlaw Star, Crest Of The Stars, FLCL, and R.O.D.
Make sure you watch at least half of Trigun before you decide to buy it, my experience was one of great frustration and boredom mounting after the first episode.
Also, watch the movies Metropolis and Akira, they are probably the two most original anime and will expand your idea of what can truly be done with the medium. They are my highest reccomendations.
If you're looking to get into comedy/romance, you have to start with Love Hina. If you don't something might just go wrong and bad things will happen, I assure you. Also, GTO is probably the best comedy I've come across, it doesn't even have the strange crap (romance, magic, etc.) that you have to deal with in most comedies, just straight-shooting, serious comedy. Heh...
Watch the first season of Lupin III (and the movie, Castle Of Cagliostro) but quit after that unless you develop an obsession.
Watch as many of the Studio Ghibli films as you can (save Fireflies, maybe).
You sound exactly like me when I started on anime so many years ago, same frustrations, same american shows.
Here's a website (mine, heh...) which will probably prove incredibly useful for your future ventures, more information than you might want.
http://elotroladodelburro.tripod.com/anime.htm
Check it out.