Build the walls
You could build the walls out of just about anything, but, due to the moist conditions, you should splurge on a handful of 2-by-4s made of cedar or other rot-resistant wood for framing, and some moisture-resistant wall board (“green board” sold for use in shower stalls).
Nail a 2-by-4 to the ceiling, fasten another to the concrete floor with a bead of construction adhesive (the kind in caulking gun tubes), and cut the studs to fit between them.
Cover the walls
Put your gypsum board on the inside surfaces first. Once the inside panels are glued and screwed in place, stuff the cavities with fiberglass insulation and cover the outsides. With all of the coverings in place, get out the aerosol foam again and shoot it into all of the cracks—especially between your new wall and the (likely) ragged edges of the old walls.
A root cellar does not need to be airtight, but the tighter it is, the more control you’ll have over the air quality and temperature. Plug as many gaps as you can.
Add the shelves
Bear in mind that lower shelves will be cooler and wetter, while higher shelves will be warmer and dryer. Arrange and space your shelves to suit the items that will likely be stored on them.
Hang a door
You can use a ready-made door if you want. Or you can make it simply from quarter-inch plywood and hang it directly on the studs. One customizing touch worth considering is to make the door in two pieces. This way you can open the top half and grab a couple of carrots without letting out the coldest, dampest air at the bottom of the root cellar.
Fasten a rod to the handle of each blast gate and run it through the wall into the basement. This way you can open and close the valves without opening the door and spilling the cold air. It also will allow you to see whether the valves are open or closed without opening the door.