I used 3/4" thick plywood. I'll protect it from the elements with paint. The box needs two sides that measure 11 3/4" deep. Their top edges need to taper: they measured 29" on the longest side and 18" on the shortest side.
The box goes together quickly with 3" screws. Even Abigail (age 7) got into the act. The bottom of the box (and the shelf) were 27 1/2" x 11 3/4".
The main assembly is ready for a back. I used 1/4" thick plywood, which is plenty strong.
The easiest way to attach the back is to tip the box onto its front. Then you can drive screws directly into the 1/4" ply and into the 3/4" plywood parts of the box.
The roof is made from a single piece of 3/4" plywood I gave it a little bit of an overhang on the front and back edges. Its overall measurements with 32"x18".
The box could might mounted in any number of ways- I decided to attach a quartet of 2"x2" "stilts" to the corners. This is because they were laying around with no other pressing purpose. I liked the height that the box was at when it was perched on my bench, so I was able to attach the legs by simply letting the box overhang the bench top and then screwing the legs to the box. A clamp helped hold the legs in place temporarily.
These doors feature "euro-style" hinges that are commonly found in kitchen cabinets. That said, if your doors don't already have hinges like these, you may want to go with a simpler style- there is a bit of a learning curve to this type of hinge. If you're looking for some old cabinet doors, I suggest checking someplace in your area that offers salvaged and reused building materials.
The doors- once I cut the plexiglas and get it in- go a long way toward protecting the books and the interior surfaces of the box; the other key weather proofer is some kind of decent roofing material.
Also: the term "Little Free Library" is an official designation. There are over 36,000 LFL's in the US alone. You can register your book box and receive a plaque which certifies it as a part of the network. That said, you are under no obligation to do so.