Mole Dojo is located in the area that looks like a stone hut, due east of the Monster Village. When you enter, you'll see a mole directly ahead of you. He's wearing a hardhat. His name is Doug. Until the story gives you a reason to need him, he does nothing. When you come here after a visit to Spinach Caverns, though, that changes.
When you talk to Doug, he's interested in Parin's plight but overburdened by a bunch of work. Since she's such a pretty lady (and all of twelve years old), he can't say 'no.' Instead, he challenges her to a competition. If she wins, he helps her. If she doesn't, he won't. Too busy to work and he's proposing a competition? Whatever.
The rules of the contest, as Doug says, are simple: whoever destroys the most rocks is the winner. The timer will count down from three, then you run around the area drilling rocks. You have to charge your drill to successfully destroy them, of course, but this is actually an easy challenge to win once you fall into the rhythm. The trick is to watch for places where rocks line up in a row. Sometimes you can hit three or four rocks at once, particularly once things really get moving. Try to work on the side of the arena opposite wherever the mole is working, since he doesn't have to charge quite as long, but never shirk away from an opportunity to hit a bunch of boulders at once.
When you win, he'll leave his cave and head to the caverns. You should follow.