It depends on your interests. What attracts you to anime? Would you like romance or action? Comedy or sadness? A long anime or a shorter one?
The most popular anime seem to be Naruto, Fairy Tail, Bleach and One Piece. (Edit: I seem to have forgotten Death Note somehow; that's really popular too) Unfortunately I have not properly seen any of these anime, so it would be difficult to tell you my opinion on it. However, they probably are popular for a reason.
My favourite anime is Wolf's Rain. It's a lovely, emotional series. As the name suggests, this is about wolves, so if you're not so fussed on animals, it probably won't be your thing.
Another good anime is AIR. This one is relativity short and focuses on humans instead, but it's still emotional like Wolf's Rain.
If you're looking for a comedy, I'm led to believe there's a lot of funny anime out there. One I found funny was DNAngel which has romance and fantasy elements as well.
If you're more interested in anime movies, I strongly recommend a Studio Ghibli movie. They have a variety of different genres and are popular among anime fans.
Hope this helps :)