We usually use "That's right" if we're agreeing with something that the other person said. For example:
"I heard that you went to Jane's party last night."
We're saying that what the other person said is "right," or correct. It sounds a little strange to say "that's right" in response to a question like "did you...?"
(By the way, the phrase "you bet" isn't really used much these days, at least in my experience. Everybody knows the phrase, and it's good to learn it, since you may hear it in movies or read it in books. I can't remember the last time I heard someone actually say it, though.)