Sweep and clean the subfloor thoroughly. Go over it with a hammer and make sure there are no raised nail heads anywhere. If there are any loose or movable spots, nail them down with your nail gun, making sure to drive the nails into the joists below. You'll be able to tell where the joists are by the location of the existing nails.
Spread carpenter's glue around the back of the first piece of plywood. Press it onto the subfloor in one corner of the room, leaving a 3/8 inch gap at the wall to allow for expansion of the wood with climate change. With your nail gun, shoot nails about every two square feet over the surface of the underlayment into the subfloor.
Lay the next plywood sheets in the same manner, at the end of the first one. Leave 1/8 inch of space between the two plywood sheets and 3/8 inch between the plywood and the wall. Lay sheets until you reach the end of the floor.
Use your tape measure, pencil and T-square to mark a line across the width of the last board in the row, marking for the space that's left (and leaving the additional 3/8 inch space at the wall). Cut it on your table saw. Set it in place with glue and nails.
Lay the second course of plywood in the same manner, but starting from the opposite wall, where the first row ended. Make sure the ends of the sheets in the second course don't line up with the sheets in the first course, and are positioned at least 2 feet away from each other. Keep a 1/8 inch space between the plywood sheets.
Cover the whole floor. Cut the pieces for the final row lengthwise on your table saw to make them fit, if necessary.