If I were you I would take the tire off again and try again. Do this several times. You might think I'm joking but I am being completely serious. Knowing how to change a tube and being efficient at it is a necessity, especially if you start doing group rides. I flatted today, about three minutes later I was back on the bike which resulted in minimum waiting for the guy I was riding with.

As far as how often flats happen, I have had approximately 10 - 15 flats so far this year. I really don't even get bummed out any more, it's just part of the sport.