You can open up a custodial account through most online brokers. It's essentally the same as the way your account is set up. They can log in to their individual accounts and view, trade whatever they want. When you withdraw money from the account you basically just have to check a box indicating that the money will directly benefit the minor. I think KATX would be a good one to start with.
My dad did basically the same thing for me when I was 13