I am a beginner to GIMP but I've done some stuff in PSE.
....I tried to use the "magic wand" / fuzzy select tool, but
A few issues there. I know what you want to do but not sure how you are going about it.
First general issues. Forget what you have done before in PS, Gimp works in its own way, it has a steep learning curve, it is not PS.
Windows users for some reason fall back on re-installing when they think the application is broken. That is very-very rarely necessary. More usually it is the Gimp profile C:\Users\your-name\.gimp-2.8 that has been screwed up. Rename that to back it up in case you ever need it again. Start up Gimp and a new default profile is created.
Deleting the background.
"quote - I tried to use the "magic wand" / fuzzy select tool, but instead of activating the section I clicked, the entire image began to fill"
That is the bit I can not understand, what do you mean by activating? A selection is normally indicated by a dotted line 'marquee' - sometimes called crawling ants.
Removing a background this way requires
1. Add an alpha channel - screenshot 1
2. Using the fuzzy select tool, click in the background.
3. Cut the selection, Edit -> Cut or ctrl-x screenshot 2
Note: Not all is selected, you can add / subtract from an initial selection by changing the fuzzy tool mode.
The gimp docs are:
Chapter 14 starting https://docs.gimp.org/en/gimp-tools-selection.html
Chapter 7 starting https://docs.gimp.org/en/gimp-using-selections.html
and a short video (7 minutes) just describing them: https://youtu.be/m-Y4G8_murs
A more usual way which you can try is, Colors -> Color-to-alpha. Default is white. Sometimes those types of clip art become semi-transparent, so it is try-and-see.
rich: www.gimp-forum.net