The best way to understand the tire changing process is to visit a VCCA member and help them change a tire!
If you get to the SF Bay Area, I'd be happy to show you.
Just short of that...
Joe and his family were out for a Sunday's drive the old Chevy truck. Suddenly, they heard a hissing sound and the right rear tire went flat. Curses!! Luckily, Joe had a spare tire mounted on the spare rim already, so all he had to do was safely jack up the offending wheel, take off the wheel clamp nuts and wheel clamps, pull the rim with the flat off, and put the spare rim on the wheel, along with the clamps and nuts. 20 minutes later, the family was back on the road to Grandma's house for Sunday dinner.
As they drove on, Joe remembered a time when he and the family had a flat in worse circumstances. It was raining and they already had had a flat that day, so the spare tire's inner tube had not been repaired yet. Dang! A second flat meant that he had to take out the inner tube and repair it.
First, since they were on a dirt road and it was raining, he had to find something solid to put under the jack to keep it from sinking into the mud. After a futile search of the area, he settled for the floor boards in the cab (and crossed his fingers that they would not break).
He found some good sized rocks to use as wheel chocks, put on the emergency brake, and jacked up the wheel with the flat. After taking off the wheel clamp nuts and the clamps themselves, he found that the outer rim was stuck on the inner rim (the felloe). Grrrrr. Then he remembered a trick he learned from his mechanic. Finding a rock that was around 5 pounds, he struck the backside of the wheel (under the truck) at the opposite side of the air inlet and voila, the stuck wheel popped off.
Next came the chore of unlatching the split rim latch, using the rim tool to compress the rim somewhat (not too much or the he'd damage (bend) the rim), use the tire irons to pry the tire off, take out the tire flap, pull the tube out, find the hole and fix it (another whole process unto itself).
Then came the fun part, putting it back together again, being careful to NOT re-puncture the inner tube, pump up the tire, and continue on the drive. He SWORE to himself that he would always carry a good space tire with air in it from now on.