Calvin has been seen using balloons on a few occasions.
Usually though, he just fills them with water to use as Water balloons to throw at Susie or at Hobbes for a practical joke. Calvin and Hobbes often use water balloons as part of their G.R.O.S.S. club to get rid of Susie (by throwing them at her.)
One instance was a Story arc in which a conventional helium balloon was somehow able to lift Calvin high into the sky. (Probably another of his imagination shemes.) The balloon went thousands of feet into the sky, then popped when the air pressure outside got too low. Calvin managed to save himself with his Transmogrifier Gun though. It suggested that Calvin flew over Chagrin Falls, Ohio (Bill Watterson's hometown) due to numerious hints Calvin says.
In a regular strip, Calvin began to blow a balloon, and his head inflated instead of the balloon. In another early strip, Calvin has another helium balloon. He climbs with it onto a stepladder (wich Hobbes was holding for him) expecting it to lift him up into the air. To his disapointment, nothing happened, to wich Hobbes suggested jumping. Calvin did this, but only for him to land hard on his face and the balloon lifting up without him.