Since my dwarves aren't reproducing, I need to retrain some of them. How do I convince, for example, a miner to lay down his pick and pick up an ax?
Simply change the dwarf's labor preferences. Select the dwarf (either on the map with v, or through the unit screen), press p to select their preferences. Press enter on a highlighted labor to turn it on or off.
In your example you would turn off mining, and turn on woodcutting (which is under the woodworking section).
Currently the only way dwarves get better at anything beside combat is to do do it. So the only way to train surgery is to do surgery.