Once your little one's piggy bank is near to overflowing, get her a savings account. Ask if you can get an account without any fees at your bank, or look for a credit union in your area. For kids who have a small balance, Capital One 360, for example, offers a child-specific online savings account with no minimum balance and no fees at capitalone360.com. Parents and children can monitor the account, although only adults may make transactions. Be sure to teach your child how to track her balance, and explain how a savings account's interest will help her money grow over time.