start with the classics:
any and all miyazaki/studio ghibli
ghost in the shell
original dragonball
sailormoon
macross
the gundam movies (not the series)
zeta gundam
captain harlock (or herlock, as matsumoto stated back in 2000)
the satoshi kon movies (perfect blue, tokyo godfathers, millennium actress, paprika)
any/all the studio pierrot magical girl series, if you can get them (creamy mami, magical emi, wedding peach, etc.)
dirty pair
cutey honey
old takahashi, namely ranma, urusei yatsura, maison ikkoku
wings of honneamiase (or, as i call it, wings of honey mayonnaise)
it's always good to start with the classics, to get a good idea of what sort of series you like. pretty much all series afterwards owe much to their look/feel/style to these shows.
if you're interested in more modern shows, and by modern, i mean anything post-1995, here's a list:
cowboy bebop
trigun
fullmetal alchemist
chobits
ah my goddess (the movie is excellent)
voices from a distant star
excel saga
azumanga daioh
ghost in the shell: stand alone complex
evangelion (with certain warnings: some people love this show, i personally don't. it did, however, change the anime industry forever. if you're over 25, you may not like it much, either. when i watched it in my mid-late 20's, all i could think of was, "if you don't stop crying, i'll give you something to cry about!" it's very angsterific. i have no patience for wangst.)
we also have all these available (except for creamy mami and magical emi because they've not been released domestically) for rent at the store (gimme anime). a lot cheaper than buying.
