C172 Pilot reported a rough landing; then after departing for another destination; landed and found the prop had been damaged on the rough landing.

Narrative

I flew to ZZZ1 from ZZZ2 in a C172; rented from the flight school I have been flying out of for nearly a decade; to practice landings at an airport I have visited many times. After working the pattern for a while; I came in for a landing and bounced on the runway in gusty conditions. I did not try to salvage that landing but applied full power; carb heat off; gradually bringing up the flaps. A peek out the window on my way up and out indicated no problem with the left main gear. At that point; I had no idea that the propeller had contacted the ground and the airplane was responding normally. I flew around the pattern again and saw that the wind had at that point died down; with the windsock pointing down the runway; so I landed; back-taxiied for departure; took a swig of water and then departed to return to ZZZ2. On the way back; the airplane seemed to fly fine; with all engine gauges reading normal. There was some light chop as temperatures were rising; and as is my usual habit I adjusted my airspeed so as not to exceed Va. The only thing I noticed on the return flight was that I needed a touch of left rudder pressure to keep the turn coordinator ball centered in level flight. I landed at ZZZ2 and while the landing was OK; I had a little difficulty controlling the airplane on rollout. When I returned to the flight school ramp and stopped the engine; I noticed to my shock that the prop tip was damaged and immediately alerted staff. I am lucky that I was able to fly back the 37 miles to ZZZ2 without incident. I should have left ZZZ1 earlier; when the wind was kicking up. I was lucky that the wind had calmed down for my final landing at ZZZ1; but I should have considered the possibility of a prop strike; which did not enter my mind. I intend to undertake remedial training with a CFI at the flight school; and also another place I rent from; until they consider my performance acceptable. I am appalled and ashamed that this happened.