The private pilot stated that he entered a left traffic pattern for runway 20 and completed all checklist items. He flared the airplane at 25 feet above the ground. The airplane touched down on the main landing gear and bounced back into the air. He applied aft pressure on the control yoke; the airplane touched back down on the runway, and bounced back into the air. He applied full power; the airplane touched back down on the nose wheel, and bounced. The nose wheel tire blew, he applied aft pressure on the control yoke, and the airplane touched back down on the runway. The pilot applied back pressure on the control yoke, increased power, taxied to the ramp and shut the airplane down. Examination of the airplane revealed the that firewall was bent and the fuselage skin was wrinkled. The pilot reported winds from 050 degrees at 9 knots.