Wood is collected when felling trees with an axe.
Felling a normal-sized tree will give you between 13 and 17 wood. After felling the tree, you can also destroy the stump which will give you between 5 and 7 wood. Be careful, as felling trees near water can result in the loss of some, or all of the wood. When farming wood it is clever to leave the stumps as they will grow back within 10 days.
Wood can be spent on a variety of different things, including: