A wood cutter also requires an available axe. The quality or material of the axe has no effect on any aspect of woodcutting - a no-quality copper axe is the same as a masterwork steel or adamantine one. (The same is not true for axes in combat.)
This skill increases rapidly. With an ample supply of trees, a dedicated wood cutter who does not do his own wood hauling can reach legendary status in well under a year, and can singlehandedly meet the lumber needs of most types of fortresses. A single dwarf can also function well as a dedicated wood cutter and plant gatherer, provided that this dwarf is not your sole food provider.
A dwarf cannot have both the Mining and Wood Cutting labors enabled, as doing so would require them to dual-wield an axe and a pick.
Woodcutters are working outdoors where they are in constant danger of being ambushed by invaders or attacked by wild animals (unless, of course, you are the proud owner of an underground tree farm), but they are also one of the few civilian professions which carry weapons with them. Thus, it can be helpful to teach them how to use their axes to defend themselves (and others) properly by making them spend a few months as axedwarfs in your military.
Setting all dwarves which must work outside to Wood Cutting can be a good idea, as it means they will carry axes with them at all times - though this also means that they will usually interrupt anything they're doing to go chop trees.
Also see Axeman