You obtain timber by chopping down trees in a forest. You need a parchment from the owner granting you access to chop down lumber from the forest. The forest will tell you who the Lord is. The lord is usually the one who owns the castle nearby, however the vassal is randomly selected.
Once you obtain permission, you will need to get a woodcutter's axe from a town's arms merchant. Equip the woodcutter's axe, then enter the forest. The trees you are allowed to chop look different, and the ground is grey around the ones you can chop. They also show a "Destructable Object" UI when near and pointing at them. You can harvest a maximum of three trees before processing them by going to one of the wood pits and turning them into timber. Doing so grants you 1 timber and 3 Prestige. You can repeat this process until you have enough timber.
If your axe breaks you can keep using it as long as you don't go into your inventory. So you should be able to use one axe per trip to the forest. This is so common that chances are that you will need a new axe for each trip.