There is not a great deal of valid training readily available for puppeteers in the United States. You can go to workshops at puppetry conventions (http://www.puppeteers.org), or study at the collegiate level (the University of Connecticut has one of the only degree programs in puppetry in the country), but mostly it's a matter of learning by trial and error (and error, and error, and more error). So how do you learn? Watch good puppetry on TV, in films, on video, in live shows, and really take note of what it is that you like about it. What are the details of the way characters move, what makes you believe in the characters? Try to be analytic in the way you observe puppetry. And do research - search the web, go to the library, talk to area puppeteers. Then try to apply the things you respect and admire about good puppetry to your own work. And practice! Practice until your arms fall off. Videotape your practice sessions then look back at them with a critical eye as to how you can improve. Then practice some more.
And most importantly, get performance experience. Do live shows at your church, or at a mall, or for schools. Do shows for your friends, or do a public access cable show in your area. Just get out there and perform. Work for other puppet companies, tour, create your own live shows, perform at birthday parties. Perform whenever, wherever and however you can, and learn by doing. GET EXPERIENCE.
There are not many opportunities for puppeteers in TV and film. It's a very specialized and competitive field, and there's only a couple dozen people who make a living at it. If you think it's something you want to do, you'd better be very, VERY good. A good puppeteer, a good actor, and a good singer. This is Jim Henson's legacy - he raised the standards for TV puppetry, and the medium now requires highly skilled, multi-talented performers. Whether or not there's ever a project that requires your skills is a matter of luck and good timing.