To help make things more clear, you basically have two choices. You can run a single 120v 20 amp circuit to your garage or you can run a 240 volt service to your garage which would then require a subpanel installation in the garage. This subpanel would then provide numerous 120 volt circuits and 240 volt circuits to supply power for your needs.
And the cable does not need to be buried below the frost line. Since you are in Chicagoland where NM-B cable is not allowed then you probably can't install UF cable either. You'll need to install PVC conduit and run THHN/THWN wires inside, and it would need to be buried 18".
Underground is the way to go here, just make sure you call Miss Dig or Diggers Hotline so they can flag all your utilities first so you don't hit anything.
I would not suggest running the conduit under the sidewalk if you have a choice. Much easier to dig up the conduit if ever need be. And all receptacles in your garage must be GFCI protected. Here's a link that'll show you how to run a single 20 amp circuit to your garage, a 240 volt circuit would be a bit different in that the conduit would terminate to a subpanel instead of a j-box.
http://www.popularmechanics.com/home...t/1275601.html