I'm trying to get into DMing and my buddy who always DMs for us offered to teach me. He wants me to create a 2 hour long adventure that he and I will run through with me as the DM.

My problem is that I don't really know what I'm doing, I've got the basic story down (it takes place in a large mushroom forest which has recently been corrupted by a mysterious force which he has to track down and confront in order to free the village within that's been under siege).

I've been looking through the Monster Manual for something to fit in the forest and thought some mutated wolves would be perfect for the setting. When I say mutant wolf, all I really mean is a modified wolf with something like a large eye that is on its side, or a tentacle or two sticking out of a leg - just stuff to make it look abnormal and out of place, in addition to being a stronger wolf that can serve as a suitable fight to kinda kick off the forest venture.

How would I go about this? I have the information of the wolf up (its stats and whatnot) but I don't really know how to go about modifying it. Is there some sort of procedure I can follow?

For a first adventure, don’t. Don’t start modifying monsters.

In time you’ll be able to judge how tough monsters are, and what to throw at your players. There is a procedure for making and modifying monsters in the Dungeon Master’s Guide, but it requires some experience and judgement to use. But you don’t have the experience to make those judgements yet. I can tell because…

A bunch of wolves are already a challenge for a first adventure. You’ll risk killing your PCs a lot if you throw stronger wolves at them.

Just use normal wolves and describe them as mutated. They don’t need to be stronger.

The Dungeon Master's Guide is your friend

Chapter 9 of the DMG covers both Modifying Monsters and Creating Monsters. That's the best place to start for what you're trying to do.

Otherwise, have fun, get creative, and be open to adjusting the monster on the fly if it's not turning out the way you expected (too difficult or not difficult enough.)

Using a third party tool like the Kobold Fight Club can also help in determining if your monster(s) are too much or too little.

Try to stay vanilla at first

But I also would recommend against too much homebrew to start with. You're learning the ropes and sticking with official monsters will be easier for you to manage with the baseline developed by WoTC.

Consider the power of descriptions over stats, here.

Don't make extra work for yourself. Whenever you're thinking of using a custom monster, flip through the Monster Manual first to see if there's anything that resembles what you want. You want a stronger wolf? Look at the Dire Wolf stat block. You can alter the description of the monster to whatever you want without needing to change any stats. Tell the party they see a wolf with tentacles growing out of its leg and a massive eye growing on its side, but use the Dire Wolf stat block.

Remember that as the DM, you are providing all of the information about the world that the players receive. If you tell them that they see a wolf-like monster with tentacles and an extra eye, then that's what they see regardless of the stat block you're using for that monster.