Besides the original tutorial battle, training can give your character a level-up. It's best to use training as frequently as possible. You can raise your power, Ki, and HP. Each mini-game has levels as you get past other levels successfully. For example, if you complete Level 1 successfully, you go immediately to Level 2. As the numbers progress, the levels get harder.
Kami
Training with Kami is the original training. Kami's training is available before King Kai's, thus you should work on the mini-games given to you to build up Goku for battles of the first half of the game. Completing the mini-games successfully will give you a level-up in different areas. The area that gets leveled up depends on the mini-game you choose.
HP
The mini-game that gets your power up is just like blocking. Try to get the marker as close to the middle as possible within the time limit. Try completing the mini-game successfully at least 6 times to see a large improvement in your HP. If the marker goes off the bar, you have to start over again.
Ki
This mini-game raises your Ki. It's the same as dodging as you'd do in a regular battle. Press the button on 
to coordinate with the arrows beside the Z. It can either be Up, Down, Left, or Right, so be ready.