I completely disagree with BMitch here. If you live in an area with sustained freezing temperatures, you should have a plastic vapor barrier between the drywall and the studs, wherever "warm meets cold" (exterior walls, top floor ceiling). You want this vapor barrier inside your insulation so the barrier is toward the "warm" side of the thermal break provided by the barrier, thus reducing the amount of condensation that forms on it.
Paper, like the backing of your insulation rolls, is NOT a vapor barrier. Neither is the insulation itself, even though its purpose is to "trap" a bubble of air and thus provide a buffer between warm and cold. Neither of these are continuous; the insulation is cut to fit inside the studs, and so air and thus moisture can still get around the paper. Moisture also soaks into paper. 6-mil plastic vapor barrier, installed properly, has no way through it; it's sealed against the studs so the drywall screws don't break the barrier, and all J-boxes are sealed with Tuck Tape or similar vapor-proof adhesion.
As for mold concerns, there shouldn't be any, even on the back side of the wall. Because the area behind the vapor barrier can still breathe, even with the insulation, there is little chance of there being enough water in the wall to damage the studs or cause mold to form on them. The big concern is the drywall, which is a sponge for any condensation that may form on it. It will trap this water and hold it for a long time, which along with the processed wood product (paper) on its skin, is a definite attraction for mold.
The moisture barrier would generally go between the furring strips and the concrete and would be attached to the wall simply by attaching the furring strips to the concrete. If you wanted to really seal the area you could run a bead of adhesive sealant between the moisture barrier and the wood.
6mil plastic is a "vapor" barrier, not a "moisture" barrier. The difference is that a moisture barrier allows air to permeate it but prevents liquid water from going through. A vapor barrier prevents water AND air being transmitted through it. Moisture barrier is used between porous materials like concrete and wood to allow the various materials to acclimate to changing temperature and humidity without exchanging liquid water (the concrete will absorb a lot of liquid water from the ground, and then pass it into the wood which is a sponge for water, which will cause the wood to rot). If vapor barrier is used, moisture will be trapped on either side of the barrier, which can cause puddles of condensation on the moister and/or warmer side of the barrier.
So, to answer your question, you should probably use a product like Tyvek or TyPar, or an older material like roofing felt, between the furring strips and the concrete. For an installation like this, you can just cut strips of the barrier to prevent direct contact between the materials, and then simply place them behind the furring strips as you attach the strips to the wall.
Best Answer
Possibly dig (or have someone else, with equipment, dig) around the outside and put in a curtain drain to lower the water table in the immediate vicinity of the garage. Or lay a barrier on the surface and add a 2 inch (or so) topcoat over it.