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.