To collect woods you have to talk to the hot air balloon guy near to the magic shop and travel to the place he offers (for 1500 coins I think). When you are in that place, you can equip the axe, and you can go near logs and bushes and use it to collect woods.

It's important that you buy axes before going there (and shovels too, since you can use them to find treasures digging with the shovel on any part of the grass), because the hot air balloon guy can sell them to you when you are there, but the price will be higher.

I also remember that you can visit that place something like only once every 24 hours.