Everyone in the family should know that Kitty must be handled gently. Cats are sensitive. You don't want to encourage rough play with a kitten because the kitten may grow up nipping and scratching to protect herself. Cats like gentle stroking, but they will decide when they've had enough. They may prefer just to sit near -- or on -- you.
Young children shouldn't be left alone with Kitty. Children should never play roughly with the cat or pull Kitty's tail. And don't let Kitty wrestle with your fingers -- that encourages her predatory instincts and teaches her to bite hands.
There's a right way to pick up a cat: Put one hand under her chest, holding her front legs with your fingers, and support her hindquarters with the other hand. A mama cat may pick up her kittens by the scruff of the neck, but people shouldn't. When you do play with her, handle her gently -- and often. That will make it easier when you need to groom her later.
You can't really train a cat in the way you train a dog, but you can discourage behavior that you don't like. If Kitty is doing something bad, say "No!" sternly. A loud clap of the hands may help. You can also drop a heavy book on the floor, then take kitty to her scratching post. Never, ever hit a cat.