1.Fruits Basket: I think this would be a good manga to start with because it's mostly a slice of life but with just one or two fantasy elements and I think the series is brilliantly written. I believe that the characters in this series are some of the best character development I've ever seen in anything.
2.Anything by Clamp: Clamp has so many wonderful series and all their art is fantastic. They cover a lot of different type of stories such as girlier titles such as the magical girl series Cardcaptor Sakura and Magic Knight Rayearth or a romantic series like Chobits. There's also series about different supernatural elements such as Xxxholic or Tokyo Babylon. There's a series about the end of the world X and I scifi series Clover then there's a series that combinds all their past work Tsubasa Reservoir Chronicle.
3.Higurashi No Naku Koro Ni: Is a great series for someone who likes psychological horror but they should be warned that it starts out a little slow but once it gets to the murder mystery/curse killing element of the story it gets really good.
4.Death Note: Is a really good series aimed towards boys and it's a great battle of wits that involves mass murder but be warned that the main character is an anti-hero at best.
5. Melncholy of Haruhi Suzmiya: This is a very odd but very good series about guy who is forced to join a club about looking for the existance of time travlers, aliens, and espers the club is ran by a girl who is a god but doesn't know and the other members are a time travler, an alien and an esper.
6. Neon Genesis Evangelion(Anime): This is probably one of the best known anime series of all time and for good reason it has a lot of good characters that are all emotional damaged and there's action in the form of gaint robots fighting aliens which makes this a mecha anime and this is certainly the best one I've seen of that type. It's a great sereis and I suggest you watch in Japanese with english subtitles because the voice of the main character in the english dub is kind of annoying and a lot of people dislike the main character anyways although he's personally my favorite character.
7. Clannad:I love this series it's a slice of life/harem series with an element or two of the supernatural but other than that it's just a study of a whole bunch of different characters who are wonderful.