On Tython I've encountered two trainers, and the second didn't have any new moves to offer me. Do all trainers teach the same moves? Or do I need to seek them all out at the right levels in order to learn everything?
All the trainers (that are appropriate for your class) will teach you the same skills. In the top right (if I remember correctly) of the training menu you can choose to display the skills you haven't learned yet or don't have the level requirements to learn yet.
All the trainers teach you the same basic skills. IE. All Sith Inquisitor trainers will be able to teach you force lightning at level 2. Multiple trainers are scattered out to make it easier to get your skills when you level up.