I know this sounds like piracy (It probably is), but I have classes that are very easy, and I want to put Oblivion on a USB, haul it around from one computer to another, and play Oblivion in my spare time.

What files do I need to make sure Oblivion works?

There's nothing piracy-ee about having the games on an external USB drive (choose SSD for best result), or an operative system for that matter. Just make sure to set the drive letter manually in Windows to a fixed one (do this on every computer, say G: for example), and then install (use the offline installer, don't use Galaxy AT ALL, less hassle). This applies ONLY if you choose to install the game other than the first time.

https://www.howtogeek.com/786583/how-to-change-a-drive-letter-on-windows-10-or-windows-11/

Some games on GOG don't need to be installed again on other computers, practically portable and you jusr run the game from it's .exe file, while others do. If it doesn't work then chances are that the computer you want to run it on don't have the necessary libraries, you just have to see what kind of error message pops up.

Haven't tested with Oblivion myself so you'd have to experiment with that. If it's a school computer there's a high chance it won't work as you normally don't have admin rights and therefore can't install anything.

I do it all the time. my gaming pc is different then my internet pc. I've had no problem with it.

Make an external drive from the USB SSD drive. Boot partition and all.
Put the game on it.
Find a rig with a USB 3.0 port.
And when you start the rig up, choose boot option during launch.
This way you won't need to bother, with other programs bloating the hardware.
Cause everything will be run from your external USB SSD drive.