How to get a free bow
You can find some weapons easily accessible even at low levels.
Reach Cavalorn and open the chest inside his hut, the combination is 1R, 2L, 3R. It contains a key. Go behind the hut and you will find a cave, kill the molerats and enter the cave, open the door with the key you obtained. You will find a short bow (damage 24, dexterity 12).
Entering the New Camp, after the entrance guarded by Jarvis, turn right and climb the two ladders. You will reach a room with two chests. The first one, on the right, contains a Rider Bow (damage 24, dexterity 12).
At the bottom of the mine, reach Viper's forge. On the left there is a lateral passage with a guard in front of it. You will find some minecrawlers, so be aware. On the ground you will find some arrows and a Bone Bow (damage 36, dexterity 18)