I am planning on hanging stone veneer on a 12 ft by 10 ft exterior wall. I will remove the siding and homosote board and get down to the studs. Would I then: hang plywood, foil backed foam insulation board, wire mesh, then the stone; or, foil backed foam insulation board, plywood, wire mesh, then the stone? I am concerned about having the weight hang off of the insulation board.

One does not "hang" a stone veneer on a wall, one supports the stone veneer on a brick shelf in the foundation, or on a steel shelf angle anchored securely to the foundation.

Be careful with the insulation that you do not create a second vapor retarder, or a vapor retarder in the wrong place in the wall. In a northern climate, there should be only one vapor retarder and it should be on the winter warm side of the wall only.

I would omit the insulation.

Stone veneer (ie. Cultured Stone) does not need an angle iron or any support, it is supported by the wire mesh. It is fine to apply on top of insulation (pending the insulation suits the application) just ensure that the insulation is not too thick and that you can penetrate the studs sufficiently with your fasteners. Go to the Cultured Stone website, they have instruction on proper installation for this application

I have heard that stone veneer is cheaper to install than real stone, but the price for the stone veneer is more expensive. Bottom line...Does anyone know which is cheaper to buy and install or are they about the same?

There are several companies that make real stone veneer. It looks much more elegant than cultured stone. In some instances cultured stone veneer is less expensive, but in many cases, not by much. And the advantages of genuine stone far outweigh those of cultured products. Typically, Veneer is going to be more expensive than full dimensional stone, but you'll save more on the installation. One company in particular that manufactures natural stone veneer, as well as natural full dimensional is Arnold Stone, Inc. The prices are extremely competetive. You can check out their website here Arnold Stone Natural Thin Stone Veneer