That's a tough question because there are few puppetry schools and no uniform job requirements. Whether you are working in theatre or film or TV matters as well.
I think the most important thing is a sense of play. A neat little exercise I have seen at puppetry auditions is to have a collection of toys or objects and then ask people auditioning to go away with them for a few minutes, then come back and tell a story with them. The idea is to test someone's ability to animate simple objects, give them character and tell a story. The theory behind that being that if someone is creative and has a wonderful sense of "play" they can be taught basic puppet manipulation skills.
On a more practical level, good skills to have include dance and movement (mime, Commedia dell'arte, etc.), singing, voice and theatre. If you are going to work in film/video/TV taking a basic course in television production at a community college is a great idea so you are familiar with the process too.