As many of you know, there is no such thing as a "puppet supply superstore". The variety of what is used to build puppets is just as vast as the number of builders. I love building custom puppets, especially when my customers have an idea of what they want their puppet to do. Then I get to figure out how to make that happen- and where to find the materials to do it. I am glad to be able to have resources for ordering materials, and also good friends who will point me in the right direction. (The Institute for Ridiculous Science helped me find the company to order this fantastic tubing!)
Where else do I get supplies? Thrift stores, hardware stores, garage sales, curbside (before the garbageman!), craft and hobby stores, and fabric companies. The main parts of my puppets I buy new, quality materials, but you'd be surprised what kind of cool accessories (glasses, hats, etc.) and props you will find at thrift stores and garage sales. Keep your mind open to opportunities and always have a sketch pad/journal along. Puppet ideas are constantly popping out at me at odd times. If I don't write them down, they usually float away. :)

There are many different things that you can use to build a puppet. A sock, a spoon, fleece, a box- the list goes on and on.
My monstery puppet will be built using a skeleton I construct from foam. The poly foam is 1" thick, and sold in rolls at craft stores for use in cushions and such things.  I just picked up a new roll today- I try to watch for 40% off coupons for my local craft stores. The glue I use is called Weldwood contact cement. 
I have built many different types of eyes. You can make them from styrofoam balls, ping pong balls, plastic spoons, ping pong balls and many other things.  The eyes I will be using for this puppet, I order from a supply store. *We'll talk more about the eyes when I am on that step :)
I end up hand sewing quite a bit of the puppet. I am pretty rough on thread, and found that a spool of upholstery thread works the best for me. It makes things sturdier, and if I need to use a lighter weight thread in some areas I can always do that.