To fell trees and make logs for your farm, you will need a Lumberjack Cabin.
Unlike most other buildings, the Lumberjack Cabin cannot be found in the shop. Instead, a derelict Lumberjack Cabin is hidden deep in the woods, to the northeast of your farm:
To unlock it, you will need to expand your farm all the way up there to include it - until it is inside your farm property. Once you have done that, you can click on it to renovate the ruins and use them.