My toilet makes a gurgling noise and shortly thereafter water comes up from my basement shower drain and the floor drain in utility room

And your question is how to unclog it?
Start with a snake in the easiest drain, possibly the floor drain, or look for a cleanout plug in the basement, or call a drain cleaning company. They have cameras nowadays.

Hi Jtubbs...

It sounds like your main drain line is partially clogged.

As Joypulv indicated, you'll need to find your main drain cleanout and snake the main drain line to remove the blockage.

Questions? Let us know, OK?

Mark