Gonna second Lucky Star and reinforce Magical Girl Lyrical Nanoha/As/StrikerS/the other half dozen spin off manga you might want to look into after watching those three as a good (and a bit more modern) example of mahou shoujo.
Personally, I fall more towards the neutral/shounen side of anime tastes; not a fan of romance/love stories, because love stories all sound the same to me so I don't enjoy them, and slice of life stuff usually just doesn't interest me (though I absolutely loved lucky star). Doesn't really say anything about the quality of the work, just that I, personally, don't enjoy them. Same could be said about people who don't like mecha series and what not.
Anyway, what I'm saying is: when you start out, make sure to sample a wide variety of stuff, a lot of them will surprise you. Case and point; pretty much anything made by CLAMP. Fair warning about CLAMP though, is that while their style is universally Bishounen/Bishoujo, and almost everything they do ends up either being pretty or adorable, they also seem to have a really dark sense of humor... This is especially true of their collaborations, like Blood C, which manages to be both incredibly adorable, and a gorey ********* at the same time. On a related side note, I loved Angelic Layer.
Also, don't let your opinions of quality be swayed by the big three (One Piece/Naruto/Bleach)
My list of recommendations, in no particular order, including various genres to give you a pool of various types to get tastes of (I really can't give you much in the way of shows like card captor, that's sooooooooooooo note my genre, but Haruhi I can do.). Most definitely not all inclusive, or this list would be even longer than it already is. Only including series I've seen all the way through, and remember well, and are completed. and not including all of them.
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Neon Genesis Evangelion (go in with an open mind)
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Tengen Toppa Gurren Lagann (from the makers of NGE, but in the completely opposite direction)
(I feel these two should be watched as a pair, NGE first, TTGL second, but NGE is mandatory. You cannot call yourself an anime fan if you haven't at least seen some of it.)
- Any one of the
good gundam series (Top 5 recommendations:
Original,
Wing,
G,
SEED,
00)
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Hellsing (the anime is good, though the manga is worlds better)
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Cowboy Bebop -
Death Note -
Magical Girl Lyrical Nanoha and continuations
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Lucky Star -
Code Geass -
.hack//SIGN -
Heroic Age (This one needs more love, it's got a really interesting story hidden behind its surface of space ship fights and giant robot-like monsters)
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Rurouni Kenshin -
Wolf's Rain (This reminds me, I need to rewatch this because it's been so long since I've seen it that I've actually forgotten most of it...)
- I could go on...