Flats are common on bicycles. It's a drag. Kevlar belted tires, Mr, Tuffy, watching where you are going, etc. all help. It sounds like you are pinching the tube while trying to get the tire back on the rim. Is this what you mean by pinch flats? If so, we can give you some tips on tire installation.

DannoXYZ is describing what is normally meant by "pinch flat" and these are usually caused by having too low pressure in addition to hitting the edge of some broken pavement or such. Use a good floor pump with guage and top them up before each ride.

I don't think you are going to like the airless tires. Very few people have any experience with them, so you may not get any answers here to questions of quality, availability, installation, etc. of airless tire systems.