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.