I am an extremely new user in Krita and probably don't know whether this is a bug or if I did something wrong, but I attempted making a new animation, had a few frames finished, and decided to save. I pressed Save multiple times. Next day, I came back to work on it and the file I saved it under was gone. I looked up tutorials, but they never say how to save animations. Am I supposed to Export it despite having no intention of uploading it anywhere (for now)? Is there somewhere special I'm supposed to save it under? I am very confused.

An animation in krita doesn't need any kind of special treatment for saving, it is saved as a .kra file (Krita document) by default. When you pressed 'Save' for the first time then the Saving As dialogue window would open to have shown the destination folder location in the top 'Look in:' bar. You can save it with any name and anywhere that your OS will let you save it to, just like any other file you work with. This destination folder location may have changed between using Save and using Open the next day if you've been doing something else on your computer that involves working in a different folder.

Are you sure that the destination folder you saved to is the same as the one that is presented when you use Open?

As a test, start krita and create a new file then File -> Save it using file name 'testkrita' and make sure that the destination folder is a 'sensible' place to save it to (i.e one of your personal data folders). Then press File -> Open and you should see testkrita.kra listed in the Open dialogue.

You could try using your OS file search facilities (*.kra) to try to find the animation you originally saved.