Easy Carpentry Level Up

One of the things carpentry requires is a set of tools. Try to get your hands on early loot like a hammer (ball-peen, stone, or regular) and a saw whether it being a garden saw or regular. This tools are what you need to start leveling up your carpentry.

Another thing you might need is a few skill books to boost exp gain. Books like Carpentry level 1 and 2 will help the most and the other books being for late game.

If you chose the occupation carpenter you are already set for making most carpentry items.

The next thing you need to do is find a couple large houses. Kill any zeds that might be lurking inside them. These houses have all the furniture you need to level up your carpentry. Before you start tearing down furniture read your skill book level 1 if you have it. Turn your game speed to x3 for faster reading if you don't want to afk.

Finally you are ready...
Start tearing down furniture in those house you looted. First right click then click "disassemble" then your character will start to hammer away at the object. Once you destroy it you will be granted exp for your carpentry skill.

After you level up to level 4 of carpentry you will have numerous things you can craft one of which being a rain collector barrel.

All recipes require a set of planks, nails/screws, and/or other wooden components. A rain collector barrel requires 4 garbage bags or a tarp.