I am curious how the process works and if I can do it myself or if I need to call a plumber.
It depends on where the clog is in the line and what is causing the clog. If it can’t be cleared with a plunger the next step is to use a snake. These are available at most of the big box home improvement stores. There are hand operated models and some with electric motors or that attach to a drill motor. They are relatively easy to use. They consist of a metal cable that extends into the drain line. The end of the cable has a tip that will break up the clog. As it is inserted it rotates either by hand cranking or by the motor. Once the clog is removed the cable is wound back up out of the line. If that does not work it usually requires the toilet to be pulled and larger equipment being used. I recommend consulting a licensed plumber at that point.