You won't receive too many replies on here as this is a UK based website, and building methods and terminology differ slightly. However the principles are all the same so I'll get you started.
Several ways it can be done, the simplest is just to fix some timber to the walls, 2x2 would be adequate, then attach your styrofoam over this (can use glues or screws) then plasterbo..sorry drywall over this, making sure the screws are long enough to go through drywall, foam and most of the way into the timber.
Vapour barrier in our climate would go directly beneath the drywall, in fact we often use stuff that has a foil backing for this purpose.
As you are insulating to keep the cooler air in, this changes the game somewhat, and you should probably be putting a poly sheet all over that concrete wall (beneath the foam), then again under the drywall (over the foam) if you can use the foil backed drywall then this speeds things up somewhat. Just remember to tape up all polythene overlaps, you don't want gaps or holes.
You can also get insulation already stuck to a sheet of drywall, if you can get this then fix that straight to the studs, or use a drywall adhesive, using what we call the 'dot and dab' technique.
You can probably attach the foam to the door, not sure about the drywall though as this may affect the doors weight/hinges/fixings.