You receive the Iron Pickaxe from Demetri when you enter the Quarry the first time. The Quarry has five levels. Each level has between 10 to 13 rocks. If you leave and return the rocks regenerate. On the lower levels you need more energy to break the rocks, but there is a better chance of finding stones. Level 3 has a table where you can eat to replenish energy. The nearby hot spring is useful because you can restore energy there once a day. If you run out, you are automatically taken back to your house to sleep.
In the quarry, all you need to do is stand next to a rock and press the action button [spacebar] to break the rocks (drops are random, haven't worked out all the ratios yet; I got as high as 9/10 on level 5 and 3/10 on level 1). The Iron Pickaxe equips automatically once you start mining.
The number of coins you earn for shipping each gem can be found on the Sell List.
To break rocks more quickly, upgrade the Pickaxe by talking to Benjamin at the Craft Shop. If you inquire about the Pickaxe on the counter, he will tell you how many more rocks you have to break. You need to break a total of 250 rocks for the upgrade. The Pickaxe can be upgraded twice.
It bears repeating that mining uses LOTS of energy!