The first bonus game will appear when you complete the Dock level (level 4). You have to stop a bulldozer from pushing you off the pier by attacking the driver's space with jump moves. Destroy it before the bulldozer gets to the end to receive a points bonus. If you fail, another character will jump in and destroy it for you, although you won't get any special bonus points for it.
This bonus game will come up after you complete level 5. A computer needs to be destroyed, but you are far away from it. You have to get to it but as you run across, barrels will drop down from above to knock you over. The simple aim here is to run the gauntlet, avoid being flattened by the dropping barrels and destroy the computer (one attack will destroy the computer). If you manage it you'll just get a load of bonus points.