How should I insulate my partially finished, currently uninsulated basement?
I have a similar setup as you.  Same Climate Zone (Georgia, USA) and similar type of basement and walls.  Since the vapor barrier could cause water condensation in the wall if you go with drywall, you'll want to remove it.  I am going to seal my wall with a heavy masonry sealer like DryLock first.  Then use foam board because it would be mold resistant.  If you use furring strips I would probably paint them too to inhibit mold growth.  I would insulate all the exterior walls.

You didn't say if you were going to refinish the area after removing the paneling, but if you do, consider waterproof/water-resistant materials like mold-resistant/mold proof drywall and PVC trim. Basements in general are notorious for water infiltration and mold damage, so this would help mitigate it.  Also, if this area is to be made into a livable, usable space where people could spend some time in, consider getting a radon test done as well.

You didn't state what material your garage door is made of.  If it is wood, I would probably just replace it with an insulated metal door.  If you have a metal door that is un-insulated, you can cut foam insulation inserts and fit them between the panels.  For looks you could also screw new sheet metal rectangles over and paint it if you don't like the foam to be visible.

Also, I'm not sure about your ceiling in the unfinished side of the basement.  You mentioned a drop ceiling.  If there is drop ceiling in that area, then use insulated drop ceiling tiles. You can make your own out of luan plywood and some thin foam insulation if you want.  if not, then you should be able to use batt insulation to insulate the ceiling joists.