The Mine in Leaf Valley is located near the Lake area, and right across from Funland's camp. Before you can start mining on your own, you'll need to work part-time for Rudolph 5 times. After that, you can dig at the mine whenever you please.
Mining in HoLV is actually pretty different than other HM games, and can take some getting used to. But you should definitely focus at least a little of your time on mining because there are certain benefits that come with it:
Money
You can increase your friendship levels with certain characters by bringing them ore/gems
Certain Event paths require you to have mined a certain number of items
You can bring gems to Zann in year 3 and have them made into earrings