How to get a free bow
You can find some weapons easily accessible even at low levels.
Cavalorn
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).
New Camp
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).
Old Mine
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)