I can't speak for all of the shows but at Dixie Stampede kids under 3 are free IF they sit in your lap and share your meal. If you want them to have their own seat, then you would need to pay the child's price for them.
Also, there are no spaces for strollers or infant carrieres inside the facility because of the multi-level nature of the arena. A very limited number of infant booster seats (under 25# I think) are available so request one when you make your reservation if you need it. They attach directly to the table and your server will place a chair underneath the booster for added security. Just remember, this does count as a seat so you will need to purchase a child's admisssion if your child needs the infant booster seat.
For older kids (3-5ish) who are vertically challenged, ask your server to stack 2 chairs to bring them closer to the table and allow them to see better. The chairs are meant to be stacked for storage so they are very stable for the kids.