Wood is a pretty sturdy resource for the Survivors to create basic buildings and some lower-level upgrades.
You can get Wood from some mission types that reward it. You can also set your Survivors about cutting down trees, fallen logs and small patches of woodland, each of which take a different amount of time to complete.
Survivors with a high build skill can cut trees faster.
By building a lumber mill, you can produce some wood every day. That is especially helpful when you cleared out most of the wood and your survivors are busy with other tasks. Build a lumberyard as early as possible!