Very well said, but I would add one recommendation:

Do not jack the tire up a couple of inches prior to loosening the nuts. It's much easier to loosen your nuts if you leave the tire on the ground and then jack it up.

Great writeup save for one slight detail:

You will want to put the nuts on with the beveled side inwards, so they push into the metal of the spare. Otherwise, there will be a slight amount of space regardless of how tight it's screwed on and the wobbeling of the road will loosen the nuts. At the extreme, this can cause the tire to come off and your car to spin, potentially messing up your seven dollar haircut quite severely.

It's also important to remember to have the handbrake firmly pulled before hoisting the wheel off the ground.
If you have a 'fake' bolt to guide the spare tyre to the right position this should be screwed in where the bolt that's highest is BEFORE you remove all the other bolts.
...and last but not least the bolts should be tightened diagonally in order to make the wheel sit as straight as possible.
Oh, and if you didn't know how to change a tire you probably also didn't know that the wheels have to be balanced individually for maximum grip, fuel economy and what have you.
Therefore you'll need to get the old tire fixed and put back on as soon as possible.

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