Practically all of my favorite anime series have influenced me in one way or another... but if I had to choose one that really stood out, then it would be Full Moon wo Sagashite. (I'm really hooked on the manga, too!)
I love how the storyline combines a lot of different genres that I like, such as drama, romance, comedy, etc. It's a shoujo series that has a lot of cute things in it, but it also doesn't overflow with them, so you don't get tired of it. There are a lot of serious and sad moments in the storyline, so it's very balanced. It's a good series to watch when you feel like crying, that's for sure. ;_;
As for the characters--they're probably my favorite part! Their personalities are really diverse. I love the main character, Mitsuki, because I feel like I can relate to her in a lot of ways... she's very innocent, but still strong-willed, and she sticks to what she believes in. I also like how complicated some of the romantic relationships between them get later on in the series--while some of them are really bittersweet, they still make it much more interesting. I just wish that some of them did work out.
Anyway, that's just my opinion! ^_^; I'm in love with this series~ ♥