How can I verify that the ground wires in my outlet boxes are grounded properly?
Connect the new receptacle with hot going to small slot, neutral to larger slot, "ground" wire to ground screw on receptacle. Turn power back on, insert a 3-prong circuit tester into the receptacle and see if the tester says you have a good ground.

Even if you don't have a good ground you can install a GFCI receptacle and get protection from any shock that a GFCI protects from (or at least some types of shocks).

If you find that the ground is not good then unless and until you fix this, just leave the new GFCI receptacle in place and put a sticker on the receptacle stating it is ungrounded. You can legally use it that way.