I first discovered anime from starter movies such as Ninja Scroll, Ghost in the Shell and Akira though never really cared to watch more until I found Lodoss Wars and Tenchi Muyo. Here are my favorite anime series:
The Record of the Lodoss Wars OVA: The first anime that truly addicted me. It was originally based on a Japanese novel that was written about the author's D&D campaigns that he and his friends played. It features the typical D&D flair, the warrior, the mage, the cleric, the elf, the dwarf, the thief, and the berserker as they adventure across the lands. It's definitely epic and has a sort of Lord of the Rings feel to it. Lodoss will always hold a special place in my collection.
Neon Genesis Evangelion: This was the second series to hook me in. It's starts out as your standard post apocalyptic robot anime, lots of great action and cities getting blown up. Then about halfway through the director pulls a David Lynch and gives you all sorts of interesting plot twists and turns, good stuff. I recommend this series to anyone but admit that it's been hyped around lately and urge new comers to go in without any expectations of what kind of anime they are watching.
Vision of Escaflowne: Great series, high quality animation and a kick ass musical score by Yoko Kano (Cowboy Bebop). This anime manages to merge fantasy and mecha anime with great results. The characters are all memorable and receive decent character development to boot. Escaflowne is another one of those epic story arc types of series that grabs you in for the long ride and I highly recommend it.
Cowboy Bebop: I saw a fan sub of this at a convention before it was released to the US and instantly fell in love. Every episode seems to pay homage to a different action movie genre (westerns, Hong Kong blood operas, alien flicks, film noir, etc). And while there are only a handful of arc episodes giving the viewer an overall plot I think it works best that way, giving each episode a sort of snapshot feel into the main characters lives. The music was done by Yoko Kano (Escaflowne and Macross Plus) which finally convinced me that she is one of the greatest anime music composers of all time.
Great Teacher Onizuka: As esuna mentioned earlier no amount of description can explain the awesomeness of GTO. I think the strongest point about this anime is that it does something different than the standard anime comedy/drama. The character of Onizuka is unforgettable, just when you think you finally have his character pegged he does something unexpected.
Other anime that I also recommend:
Read or Die OVA (my most recent anime, I can't wait to see the TV series)
Excel Saga (hilarious, highly recommended)
Ah! My Goddess (good all round romantic/comedy)
Golden Boy (also hilarious, possibly my second favorite comedy anime)
The Irresponsible Captain Tylor (this series pleasantly surprised me, I expected less)
Trigun (western/scifi, this formula is why Firefly was so good)
Tenchi Muyo OVA (the origin of term "Tenchi syndrome", girls just can't get enough of the main character)
Macross (classic)
Ranma 1/2 (funny series, many hijinks to be had)
Martian Successor Nadesico (a great spoof on robot/space anime)
The Slayers (classic D&D funny)
Jubei Chan (comedy spin on the legendary Jubei Kibagami, great fights)
Those who Hunt Elves (to get home the main characters have to strip every elf they come in contact with? SOLD)
Perfect Blue (creepy indeed)
Urusei Yatsura (classic Rumiko Takahashi)
Jungle wa Itsumo Hare Nochi Goo (hilarity and whacky hijinks ahoy!)
Jin-Roh (a good solid movie)
FLCL (from the people who did Evangelion and Wings of Honneamise)
Love Hina (a bit clichéd with Tenchi syndrome, still great regardless)
Ninja Scroll (great starter anime for any friend)
Ghost in the Shell (more great starter anime for any friend)
Spirited Away (loved this movie, actually a good dub too....so rare)
Edit: For years I've known about the greatness of Grave of the Fireflies. I was always too scared to watch it because almost EVERYONE man or woman that I've talked to who has seen it ended up crying near the end.