The tire was damaged as well as the tube. I took it to my LBS, and yes, they said the tube must have gotten pinched under the tire bead. It really made a loud noise and I am very glad that did not happen while I was riding.

So, I watched him put on the new tire. And for anyone who hasn't changed a tire yet, you need to slightly inflate the tube before inserting, and after putting on the tube and tire, inspect both sides of the rim and be sure that you do not see any black tube sticking out. Then inflate half way and check again before going to full pressure.

So, the end of this story is that I got 2 new tires, because when I asked if there were any faster/better tires, he suggested that I get 25cc instead of the 28cc that I had, and both tires needed to match.