Skill level increases the speed of mining and also increases the chance of leaving behind stone, ore or gems depending on the material mined. High skill can actually be undesirable, since excess stone can be a nuisance. However, miners gain skill extremely quickly so some amount of stone is unavoidable. Miners are required to dig tunnels, channels, stairs, and ramps, as well as remove said stairs and ramps. Miners require a pick to do any work.
In order for any mining to be done the Designations menu must be used.
If valuable gems or ores are being mined, it is best to use a Legendary miner. Burrows can be used to restrict a certain mining expedition to only certain dwarves.
 Mining behaviour
If a large area is designated for mining, miners will all tend to clear out the left side first, working from the top down as much as possible. This is not an absolute, but a strong tendency.
Once mining, a dwarf will automatically continue on to mine any connected tiles (on the same level) designated for mining, even diagonally, rather than shift to mine a discontinuous tile.
Dwarves do not tend to see tiles on different levels as "connected" (for instance, an up-stair below a down-stair), and will tend to clear one level before moving on to another, even to the point of moving far across a map to do so. They seem to mine top levels over lower ones.
If no mining-designated tile is connected, they will then shift, seeking to mine a top-left designated tile on a top-most level.