U can buy pickaxes in Bastok Port at Galvins Gear.(it's marked on the map) And i'm sure in the Bastok Mine District at Boytz's Knickknacks. (also marked on map) Hope this helps.
There are certain areas called "Mining Point"s that pop like monsters. They are always located on the wall, and in different areas. In order to find them you just have to either camp one of the spawn points or just wander through the mines systematically until you find one. However, since the points only lasts about 3-4 uses per spawn and other players will be mining too, there are times when it can be a pain just to find one. Generally the busier the area is, the less likely you'll find a Mining Point.