Rooms above garages are often done improperly. It's amazing, because it really isn't all that difficult, but a lot of people seem to mess it up.
Unfortunately, there's no "easy" way: You're going to have to remove some drywall to inspect and see how the insulation in the ceiling is done. There may or may not be a gap in the space, depending on how the ceiling was done. Even just taking the temperature above the insulation in that space will tell you a lot (above the insulation, it should be close to room temperature).
Drafts in that space are absolutely killer, it means there is a direct source of outside air. Black on the insulation is a good sign of airflow, indicating drafts.
Basically, if the ceiling space isn't totally sealed, the only thing you can do is turn the garage into a conditioned space (very expensive, in terms of both construction costs and ongoing energy costs), and even then, if the problem is between the insulation and sub-floor, it won't help.
What should be there at a minimum is a continuous vapour barrier, with a layer of insulation on top. The vapour barrier should be sealed to the subfloor or the vapour barrier from the walls above. The insulation should go all the way around the edge (in the headers), so the insulation is continuous from the floors up to the subfloor. There should absolutely be no drafts, exposed concrete, or anything directly connected to the outside that is uninsulated.
The best way (in my opinion) to seal this space is to use closed-cell spray foam, and form a continuous seal across the bottom of the entire subfloor. This gives you great insulation, no drafts, and acts as a vapour barrier as well (vapour barrier is not needed in this case -- though some building inspectors still don't understand this, so check your local codes). It also is better at ensuring fumes from vehicles in the garage can't get into your living space (in theory, vapour barrier prevents this as well, but 6mil vapour barrier is easier to puncture than a couple inches of hardened foam).
Though you may be able to find something obvious and fix it, be prepared that there's a possibility that the only "fix" is going to be to completely tear down the ceiling of the garage and re-do it properly.
If you're going to spend money and time on this, do it right.
The easiest way to insulate concrete block is to glue foam insulation board directly to the block. These boards come in 4 ft X 8 ft sheets and different R values. The other method is to build a wood or steel stud wall along the block to hold blanket style insulation along with electrical wiring etc. This method is a lot more work, but gives you a base to install sheetrock later on.
Depending on whether or not you want the attic area insulated or not, blanket or blown in insulation can be installed into the ceiling. ( this assumes you have a sheathed ceiling). Otherwise, you can install blanket insulation between the rafters. Depending on budget, sprayed in foam insulation is fantastic. You may also want to look at some venting to help exhaust hot air during the summer months. I may add a few comments later, but here's a start.
Best Answer
Chicago - Cold. Vapor barrier to inside. Moisture concern solved.
Half-concrete wall - if you only insulate the frame part, the concrete part will still be wicked cold. You might want to consider insulating the frame part and then covering the whole wall (concrete part and stud bays) with rigid foam sheets, taped or sealed as the vapor barrier, and then drywall - which would give you some insulation over the concrete part.