I've tried this before, and I used one of those pre-fabricated hardwood tabletops from a Big Box store. It worked ok in the end, though it was an expensive solution. The thing that tripped me up the most was the difficulty I had in finding the exact center so I knew where to bore the hole for the shaft.

This time I want to cut the wheel myself, so I know EXACTLY where the center is. I'm thinking that 3/4" or 1" plywood should do the trick.

Screw a strip of wood a few inches wide by ~2 feet long to the base of your router, with the router at one end of the board. Drill your shaft hole in the plywood, and drill the same size hole in the board attached to the router, with the hole centered 18" from the inside of the bit. Using a dowel that matches your shaft size as a center pivot, you can now swing the router in a neat circle around the center bore.

Of course this is only economical if you already have (or plan to buy) a router. If you don't, you can use some basic geometric construction techniques to find the center of an existing circle