The Anatomy of a Toilet
Using a utility knife, gently cut between the toilet and floor. Remove the toilet by carefully rocking it back and forth until it comes loose, (remember, there is a little water in there!) and place it on the old towels you set up earlier. Don't drop it! Then, using a putty knife, remove the gross wax gasket that's been under there for years and throw it away. Place a rag or wad of paper towels in the drain pipe, just in case.
Ready for the New Toilet!
There are several different types of wax rings to choose from, however in most situations, wax gaskets without plastic inserts should do the job just fine. Insert a new, not yet gross, wax gasket onto the new toilet bowl, with the tapered side facing away from the bowl. As you do this, make sure the toilet flange is in good condition and not corroded. The flange ensures a tight connection between the waste pipe and the toilet, so now would be a good time to update that if needed.
Ensure that the toilet is now level on the floor. We don't need to get into what might happen if you install a crooked toilet...
Fill the cap covers with plumber's putty and place over the bolts.