At what age is it safe for my baby to go on kids' rides at amusement parks?

Maturity. When they are tall enough and mature enough, but only you and your child can evaluate this together. Start off slowly and go on a basic ride together, then advance slowly to the child's comfort level.
Depends. The safest way to approach this is to use some common sense and to make sure that your child meets the height and weight requirements for a specific ride. If you think it may not be a good idea, don't do it.