This repository has a bunch of resources to get you familiar with Mathematica. The main file is cheat_sheet.nb which is a Mathematica notebook that acts as a guide starting off with the basics and building up to more complicated function calls. There's a pdf version cheat_sheet.pdf, but I recommend using the Mathematica notebook version since it's interactive and you can run the cells!
There are example problems available in example_problems.pdf to try out and become more familiar with Mathematica. These are all problems I've come across in my classes (except for maybe a few I made up on my own), so it should be clear how useful Mathematica can be in your courses. The actual answers are also given in that document, but if you get stuck problem_solutions.nb has a the code broken into nice steps showing how I got to those answers. There's a wide range of math levels for the problems, so skip around to the ones you think are most useful for you.
Finally, there's far too much to cover since Mathematica has tons of functions and constantly adding more. There are additional topics in the Advanced_Topics folder that cover topics from Vector Calculus to making animations.