For a beginner, I recommend the anime from Studio Ghibli especially the ones made by Hayao Miyazaki. In 2003, we happened to be visiting the Ghibli Museum in Tokyo when Seto was interviewed for an International Herald Tribune article.
Here is the list of movies from Studio Ghibli that I had watched: Princess Mononoke, Totoro, Nausicaa, Grave (Tombstone) of the Fireflies, Laputa, Kiki Delivery Service, Porcoros, Panpoco, Whisper of the Heart, Only Yesterday, Spirited Away, The Cat Returns.
The first anime I watched was “Princess Mononoke” by Yoji Matsuda from the famous Ghibli Studio in Japan. It was a great introduction to the world of anime. Blended into the cartoon were issues about man’s co-existence with nature, the spirit and importance of nature, the good and bad sides of human beings etc. To introduce such serious ideas into a cartoon and to be able to carry it off is amazing.
Learn more about Studio Ghibli from wikipedia. There are also several good fan sites including nausicaa.net and onlineghibli.