Wrap the house in a vapor barrier or moisture membrane, such as polystyrene film, stapled over the wall sheathing with a construction stapler. Tape the joints and edges of the house wrap at doors, windows and other openings. Cut out the wrap at openings with a utility knife.
Mark a line 3/4 inch above the bottom of the sheathing with a tape measure and snap a level chalk line at that height all the way around the house. Install the first foam board panel at the corner of a wall. Put the bottom of the panel on the chalk line and use a straight board to align it at the edge; use a framing square to make sure that corner is square.
Nail the panel in place with 3/8-inch head galvanized roofing nails long enough to penetrate the foam, the sheathing and into the wall studs at least 3/4 inch. Drive the nails with a hammer but don't pound them into the foam board. Leave the nail heads flush with the surface of the foam. Put nails 12 inches apart on the edges, 16 inches apart in the interior of the panel.
Set panels vertically, to extend from the bottom of the sheathing to the roof juncture. Use 9-foot panels if necessary to cover that length without having to cut and piece foam board. Butt the joints of foam panels tightly together unless they're tongue and groove type, with a protrusion on one panel that fits into a recess on the adjoining panel.
Cover all walls with foam board panels. Overlap panels at corners, so the edge of one panel goes across the end of the other. Use a utility knife, drywall saw or reciprocal saw to cut out openings for doors, windows and other elements after the foam panels are fastened.
Tape the seams with a tape to match the type of foam board. Use tape from the same manufacturer. Seal seams and all nail heads completely. Tape openings and edges, too, to completely seal the foam panels so no air can penetrate the insulation.
Install siding according to the type. Nail fastening strips to studs through the foam board for vinyl siding. Nail wood or cement fiber siding directly over the foam board, with nails that penetrate the foam, the sheathing and into the studs. Use kiln-dried wooden siding to reduce potential moisture problems. Prime the back side of the wood with paint to further seal it.