If the garage is going to remain a garage, I would insulate the floor of the new room above. Insulating the floors would save energy over insulating the garage walls below. The big disadvantage to insulating the garage walls is that the average temperature of the garage would be higher in winter, which is bad for any cars stored in it.

If you park a car that has been driven on slushy, salted streets and park it in a warm garage, you increase the corrosion activity. It is best for the vehicle to be in a cold, ventilated garage.