A firewood shed is important for keeping firewood dry, but it isn't necessarily something that you want to spend a huge amount of money on. After all, it's going to have firewood tossed into it, so it's more important that it be sturdy than pretty. You can even make it reasonably pretty if you want to put some extra work into it.
Level the piece of ground where you will be putting the shed. Lay pallets down to create the desired size floor of your shed, and make them sturdy and flat by shimming underneath them with flat stones.
Lay a piece of plywood over the palettes if you want a solid floor. This isn't absolutely necessary for a firewood shed, but it will make it easier to walk on. Secure the plywood down onto the pallets by screwing through and into the pallets every 12 inches or so.
Stand two pallets on their sides to create a right angle at one corner of the floor. Secure the two pallets together at the corner by driving screws through them. Secure the two pallets to the floor by driving screws down through their bottom edges and into the floor. By using the same sized pallets that you used for the floor, you can fit them together on top of the floor without having to cut them.
Set pallets on edge around the perimeter of the floor to create three walls, leaving the fourth side open.
Disassemble a pallet so that you can use the boards from it as cleats between the two tiers of the wall. In order to make the shed tall enough, you will add a second tier of vertically positioned pallets on top of the first tier to make the walls two pallets tall.
Screw one of the boards from the disassembled pallet vertically onto the wall on the inside of each corner of the shed so that they extend above the top edge of the first tier of pallets. Screw additional boards along the walls as necessary to provide a surface into which you can screw the second tier of pallets.
Put pallets on top of the first tier of pallets, and screw them into the boards that you have attached. If necessary, add more boards to the outside of the shed to make the two tiers sturdy and immobile.
Lay 2-by-4s horizontally across the top edge of the top tier of pallets. Screw the 2x4s into the pallets to secure them.
Lay pallets horizontally on top of the 2-by-4s to create a roof. Lay plywood over these pallets and tar paper over the plywood to create a waterproof surface. Secure the tar paper to the plywood with roofing nails.