Even though I'm late to the thread, I'll pitch in some of my suggestions:
1) Adknowledge Osamu Tezuka and watch/read one of his works: Tezuka is considered by many to be father of modern anime and manga and his works have influenced many artists and creators. His most well-known work is Astro Boy, which spawned 3 television shows (one in the 60s, another in the 80s, and the last in 2003; all are on DVD and the 80s and 2003 series are on iTunes) and an upcomming CG movie from the studio behind TMNT. If you're near a Barnes and Noble or major bookstore, read some of Tezuka's manga as they span numerous genres and ways of storytelling. On iTunes, you can buy some of Tezuka's animated short films for $1.99 each; I recommend "Jumping" and "Broken Down Film" because they don't look like or act like typical anime.
2) Watch Macross Plus: This was my jumping-on anime back in 1995 when I watched this on tape. Even though it's based off an established franchise (Macross, which became the first 1/3 of 80s franchise Robotech), you don't really need to know Macross to enjoy Macross Plus. Set in the year 2040 on a planet colonized by Earthlings, it involves 2 former childhood frienRAB who become bitter rival test pilots for experimental transforming jet fighters. Thing get even more complicated when another childhood friend returns home, but it's because she's the producer for a virtual idol and is performing on the planet. It has elements of Top Gun and does carry some intense drama. The animation can still hold up to this day, the music's awesome, and the English voice cast features Bryan Cranston, the dad from "Malcom in the Middle", who went by another name for M+. It's 4 episodes long, each episode about 40 minutes in length, and is only $6.99 for all 4 on iTunes. A movie-length version (only available in the original Japanese audio with English subtitles) is also available to rent off of Xbox Live Marketplace.
3) Watch the Ghibli Movies: The movies of Studio Ghibli have been mentioned frequently on this thread. Most of them have been released in the States by Disney, who gave them all-star voice casts, excellent translations, and care in the overall product. Also, check out The Castle of Cagliostro. It's directed by Studio Ghibli founder Hayao Miyazaki (who also directed most of the Ghibli library and won an Oscar for Spirited Away). Steven Spielberg once called it the best adventure movie he ever seen.
4) If it looks good, give it a shot: If anything meets your interest, just watch it or read it. If you're into sports, try Slam Dunk, Real (both basketball), Eyeshield 21 (American football), or One Pound Gospel (boxing). For comedies, there's Shin-chan and Love Hina. I also enjoyed Ah! My Goddess (or Oh My Goddess! as the original manga is called), Samurai Champloo, and Paprika.
Hopefully, you'll find what you like and stick with it.
