You obtain Timber by chopping down trees in a Forest. You need a parchment granting you access to chop down lumber from the Lord's forest who "owns" it. The forest will tell you who the Lord is you need permission from in order to chop trees. The Lord is usually the one who owns the castle nearby, however which vassal it is is random per playthrough. This is purchased from the Lord so you must find him.
Once you obtain permission, you will need to get a Woodcutter's axe from a City's Arms Merchant (at marketplace). Equip the woodcutter's axe, then enter the forest. The trees you need to chop will look different than the rest (look for the white trees to orient yourself towards the direction you need to go). There are both white and brown trees you can chop down, and the ground is grey around the ones you can chop. 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 3 prestige, aswell as giving you 1 timber. At that point you can continue harvesting new trees.
You will know you're at the right tree when you see "Destructible" pop up when you are near the screen, and when you chop you will hear it connect (versus the whif sound when you hit a wall) and the health bar will pop up showing the damage done to the tree.
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.
You may want to take one axe per trip though since if you have two axes in your inventory and break two axes in a single trip you'll lose them both.