Some other action stuff worth mentioning: (Grouped in priority first to last.)
Bubblegum Crisis 2040
Kaze no Stigma
Read or DIE!
If you want to try some high school slice of life stuff
is the place to start.
, of course.
Great Teacher Onizuka
are some other really good ones.
If you want something that pulls at the heart strings,
and Kimi Ga
are probably the best.
There's also Full Moon wo Sagashite
it's really girly, but I dare you not to cry. And the music is awesome.
Ah! My Goddess!: The harem anime version of Bewitched! Watch if you want to laugh and say Awww! every once in a while.
Seconded. Of course it can't be mentioned without the obligatory Maison Ikkoku, Ai Yori Aoshi,
and Love Hina
And the other romcoms like:
Midori no Hibi
I'm sure I'm forgetting something, and a lot of these are pretty old as I really haven't watched anime much in the last five years or so. But it's a start.