First off, I recommend getting a DVD player first, so you won't have to fight yourself over picking dub or sub. Plus your anime will last longer and won't get those annoying track lines.
Depends on your tastes and the kind of content you can stand.
Lupin is an old series but a good watch, it's one of those series where there's a lot of filler so it doesn't matter where you start as long as you watch the first episode to understand. Some parts contain graphic sexuality just to warn you.
If you liked Tokyo Movie Shinsha's animation with Tiny Toons or Animaniacs, then I also recommend getting or renting the Lupin movies such as Castle of Caligstro, Dragon of Doom, Fujiko's Unlucky Days or First Contact (sorry, I don't know if all these are domestic).
AIC's offerings can be classic, such as the original Bubblegum Crisis, Moldiver, Tenchi Muyo or Burn Up (not Warrior, Excess or Scramble, just the original). Mostly due to the visual style and good usage of color.
Series based on Rumiko Takahashi's manga can run for a long time and I'd only recommend finishing if you're that devoted. These series are Urusei Yatsura if you're into 80s vintage, Ranma 1/2 if you want sex comedy or Inuyasha if you want something epic. Of course there's also Rumic Theatre, Mermaid's Forest and Maison Ikkoku.
Also if you like fantasy or medieval settings, there's "Those who Hunt Elves", Lodoss War, Slayers, Berserk and the OVA adaptions of the video games Fire Emblem (Nintendo/Intelligent Systems) Ys and Legend of the Heroes (both Nihon Falcom). There's also Orphen but I got bored of that series quickly.
Actually, I'll make a list. Most of this stuff is licensed or stuff I haven't watched but was interested in watching and ranges from age group.
Azumanga Daioh (comedy)
Excel Saga (heppoko or parody)
Steel Angel Kurumi (historic fiction/comedy)
Please Save My Earth (drama/fantasy)
Full Metal Panic (action/comedy/drama)
Gundam 0080: War in the Pocket (drama)
Irresponsible Captain Tylor (sci-fi)
Naruto (shounen or adventure with young males in mind)
KareKano (licensed as 'His and Her Circumstances', a romance drama)
Jungle wa Itsumo Hale Nochi Guu (comedy)
Scryed (sci-fi)
Love Hina (sex comedy/drama)
Neon Genesis Evangelion (sci-fi/drama)
Pretty Sammy/Magical Project S (Magical Girl)
GeneShaft (sci-fi)
Argentosoma (sci-fi)
Battle Athletes (Sci-fi/Sports)
Lain (Technological Drama)
Niea_7 (sci-fi/comic drama)
Tsukikage Ran (historic fiction/drama/comedy)
Rurouni Kenshin (historic fiction/drama/action)
Spirit of Wonder (historic fiction/drama)
Aoi Yori Aoshi (romance comedy/drama)
Hana Yori Dango (romance comedy/drama)
If you have bittorrent and a broadband connection, here are some fansubs of series I recommend
Aishiteruze Baby
Keroro Gunsou (Sergeant Frog)
Midori no Hibi
Uninhabited Planet Survive
Kenran Butoh Sai: The Mars Daybreak
Konomimi