How can I take a fairy tale and turn it into a satirical piece, using all four methods of satire-exaggeration, reversal, incongruity, or parody?

Because fairy tales are, generally speaking, already larger than life, you may want to avoid choosing exaggeration as your main method of satire when you create your satirical piece. Of course you can still include it (caricature exaggeration might work well for a character that doesn't already have an extremely exaggerated trait), but I would suggest putting your main focus on parody, incongruity, or reversal.
Think of fairy tales you liked when you were younger, ones you know well. There are the obvious ones, like Cinderella, Little Red Riding Hood, Hansel and Gretel, and Goldilocks and the Three Bears. Then, there are stories that aren't told quite as often, like Rumplestilskin, The Frog Prince, and The Wild Swans. Because you will be presenting before a class, and you want your peers to understand the changes you've made to the story, you probably don't wont to go with something too obscure.
When choosing a fairy tale, think of what makes it conducive to satire. What changes can you make? Maybe Cinderella chooses not to end up with the prince for some reason. Maybe Little Red Riding Hood takes out a restraining order out on the Big Bad Wolf. Maybe we see a story from the villain's perspective. Maybe you bring a story into a different time period, changing the context. Just thinking of one effective change you can make to a story will lead you to plenty of ideas that make your piece a successful satire.
If you want some good examples of popular fairy tales that satirize the genre, you could take a look at Shrek or The Princess Bride.