Sometimes you can use a blow-in product provided there is no blocking between the studs or you will end up with a lot of holes. If your garage could be emptied out, it'd be way easier to just redo the drywall and insulate it properly - with a little help you could do it in a weekend.

You are right, it will help your heating and cooling. A friend of mine has a garage that's unfinished inside, and it's an oven right now. That has to have an effect on the house.

Blown-in insulation will involve drilling 2" holes every 16" near the ceiling and also just below the firestops (about chair rail height).
Unless your garage is heated and or cooled, in which case you'd also want an insullated garage door, I don't think you'll see a lot of difference in how hard the house unit is working.
Increasing the R-Factor of the insullation between the garage ceiling and the room(s) above it may give you the better result (especially if it is now only minimal).
Ventillating the garage in the summer will also reduce the 'oven effect'.
JMHO

My garage has no provisions for conditioning the air. It does abut one wall of the house which is insulated to R-13 or better. So the garage acts as a buffer to keep the wind from that wall of the house, at the least. I had lots of R-11 from the 1970's still in never-used rolls that I used for the garage walls and used R-19 in the ceiling, then drywall for the entire interior. The garage stays warmer than outside in the Winter and a little cooler than outside in the Summer. Door open time is kept to a minimum. I think the insulation is very helpful.

In insulating your garage will help mediate the temperature within, protecting whatever is inside from wide variances of temperature.