How to change a tyre on a bike
First, let all the residual air out of the inner tube, then insert 2 tyre lever thingees and kind of unhinge the tyre from the rim. Once you have it loose'ish, then get the rest of the tyre off and take out the inner tube. Feel around the tyre for remnants of what may have caused the puncture in the first place. In my instance, it was a tiny thin sliver of wire. If you don't do this, as soon as you get your fresh tube in, you're likely to get another puncture. Put the new inner tube in the tyre (off the bike), pump it up a little, then put the tyre and inner tube onto the rim again. Now folks THIS is the part that the bike shop guy made look REALLY easy, but what I'm guessing that when I'm stranded miles from home on a busy road with school pick up time looming, is going to be a smidge difficult. I'm just saying.
Then pump up the tyre till it's nice and hard, then put the valve cover thing back on. And you're on your way!