As a kid go to the bombchu game room in Hyrule Town Market. Play a second time, if you win, you will recieve a bomb bag that can hold 30 bombs.

If you're not very good at bombachu bowling, here's another way to get a bigger bomb bag. As young Link, go to Goron City. There is a large Goron rolling around. Stop him with a bomb, and he will give you a bigger bomb bag.