I couldn't find a publish button or anything similar, and I'm planning on uploading what I make to the internet so people can access my stories straight through the browser- is there any way I can do that?

I should mention that I'm doing this for my thesis, and it's likely that non-techno savvy people will be reviewing it. It'd definitely be a plus if they could just plug and play, instead of having to download the quest app and install it.
Any help would be appreciated.

First, open your file in Quest like you would normally when you're editing. Go to the tools drop down menu and select, "Publish." That will create a .quest file, which you will need next.
Go the textadventures.co.uk website, click "Create" from the menu at the top, and select "submit a game." From there, you can upload your .quest file. Since anything that is made public needs to be reviewed first, and since you would probably just give whoever needs it a link anyway, you should probably make your game "unlisted," which will make your game invisible to the general public. Then just use the link to get to it, and you should be able to run it from the web player as normal.