Though iTunes is mostly a store for purchasing music, TV shows, and movies, there are many free resources available that may be fun for your child to download and view, from educational programs on iTunes University to Sesame Street podcasts to weekly song picks. With a free, no-money-attached account, you can download these to your computer or your child's and enjoy them without having to pay for a gift card or share credit-card information.
Alternatively, you can use this free account method to set up an account for your child without allowing unlimited credit-card access, and let your child earn money to spend there through gift cards and allowances. Your child will not be able to buy anything with this account until there is money deposited or a credit card attached. If you want money available from the start, go to the step-by-steps on setting up an account with a gift card or allowance.
To start with your free, no-money account, go to the iTunes store, making sure you are logged out of all existing accounts. (If you don't have iTunes on the computer you're using, download it from the Apple site.) Find your way to the App Store, and look for the listing of free apps -- there may be a listing in the lower right-hand corner of the storefront, or click on "Great Free Apps" in the App Store Quick Links. (Don't worry if you don't have any device that can use the apps -- the set-up will still work.)
Note: iTunes frequently changes its layouts, and the pages you visit as you go through this tutorial may not look exactly like the pictures and set-ups here. However, the basic steps should still work.