We seem to have scared the poor OP away
Let me make some suggestions for different audience groups since we don't really know who the intended audience might be.
Mononoke Hime is probably the most thought-provoking work I've seen, and one of the most beautiful. I'd recommend this to anyone over the age of twelve who wants to see what anime can be. Few "villains" are more complex than the Lady Eboshi, and few couples more appealing than San and Ashitaka.
For younger girls, other than the Miyazaki films already mentioned, I might suggest
Angelic Layer. It's a bit long at 26 episodes, but there are lots of cliffhangers. Misaki isn't necessarily my favorite in the plucky girl category (Shion of the age-inappropriate
Shion no Ou probably tops my current list), but she's a decent heroine for this age group. For kids who've watched
Pokemon, a show about a girl who grows by learning how to battle in an abstract arena won't be that unfamiliar.
Mahou Shoujo Tai Arusu might be another option. It starts as a retelling of the
Alice in Wonderland story so it would probably seem familiar at the outset. It's also presented in a format for younger viewers, with 10-minute episodes, and plot summaries at the beginning of each one. It's not a format that appeals to me as an adult, but it makes sense in a show intended for 8-10 year olds. It's also one of the most beautifully-animated shows I've seen.
Somebody else needs to contribute on shows for younger boys. I liked
Hikaru no Go, but at 70+ episodes it's a non-starter for anime newcomers. I'm sure there are lots of good choices in this category, but it's not one of my genres.
For teens of both genders looking for a comedy, I'd suggest
School Rumble. The show is hilariously funny pretty much from the outset, so you can enjoy the show without making as big a commitment as a 26-episode series might seem. It's certainly the sort of show where you can watch blocks of episodes over a long period. It has enough continuity to keep you interested, but many episodes stand on their own.
For older teens and young adults looking for an "action" show, I'd also endorse
Black Lagoon. It has more in common with Western action movies than most anime, so it might seem more familiar to an audience of anime newcomers. Yet it still operates within anime conventions and conveys some feeling for Japanese culture though Rock's story.
Black Lagoon is also presented in short story arcs, so it's easy to decide whether to quit or keep watching.
For mature viewers of both genders, I'd suggest
Bartender (unless they're tee-totallers). The stories all concern love, loss, and renewal, which most older people can relate to, and the presentation is largely episodic.
Bartender also has a wonderful didactic component as well. I enjoyed learning how the Margarita was invented, and how Scotch was the "liquor of rebellion." Each show even includes a cocktail recipe and a live-action demonstration. The smoky-bar jazz piano of Ootake Kaoruku completes the atmosphere.