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Step 1: Materials/Techniques
You can use either EVA Y-20 foam (ethylene vinyl acetate) or XLPE (Cross-linked polyethylene foam). Both are similar to yoga mat material, suitable for use with hot glue or contact cement, come in a big range of thicknesses for stiffness, and after sealing can be painted if needed. For the strongest permanent bond always use contact cement. Follow the instructions and allow it to dry to a dull finish for ~15 minutes to get maximum grab.
You can cut the foam rubber one of three ways depending on the type of cut:
Scissors: Better on thick material but may leave a rough serrated edge.
Long box cutter: gives a clean edge.
Hot knife: Works well on thin material with very tight contours.
I've price shopped the entire internet. Cheapest foam: http://www.thefoamfactory.com/closedcellfoam/cross-linked-polyethylene-foam.html. My local fabric store, Joann's Fabrics, always has a weekly 50% off coupon that cuts cost on the fur which is $20/yrd.