Axiom

These are listed on craigslist for $35 for the pair, and the seller said he bought them for $120. If I decided to swap the Bontrager H2 Plus w/puncture resistant belt, 700x35c that come with the Trek, would that be an upgrade or a downgrade? And would they fit? I wanted to throw skinnier tires on there for a while, and I finally found a deal that attracted me.

700x35c Bontrager H2 Plus w/puncture resistant belt
vs.
700x25 Specialized Armadillo Urban Touring Tires

I don't think they will fit by the looks of it, but I am not sure. The only reason I ask is because they
are labeled as touring tires.

TRUMPHENT

If the new tires fit, it will be both up and down. You'll be getting nominally better flat protection. Your ride quality will take a dramatic down turn. The clincher is the width of the rim itself. I defer to the wisdom of Sheldon. https://sheldonbrown.com/tire-sizing.html 700c rims do come in a wide range of widths. Sorry, but you have to remove a tire from a rim in order to measure it yourself. I had thought about going to 25mm tires onto my 29r rims. I settled for 28mm and that's about as narrow a tire as my rims will take. The rims on my bike are 27mm wide on the inside of flange. I like this because there is virtually no sidewall bulging out beyond the rim. I have been using Panasonic RibMo tires for a couple of years now. I moved from 32mm Ribmos to 28mm Ribmos. The ride quality was definitely worse. That's the physics lesson in all of this. The larger the inner tube, the more suspension travel you have.

Seve

If you check the exterior of your current wheels there may be a label on them that denotes the interior rim width. Not all do , but at the same time a lot of them are labelled and might save you some time and then you can follow the guide that TRUMPET linked you to on Sheldon Brown's site.

e.g.
622x17
622x15C

I suspect that they will fit and you can always measure the rim yourself if there is no labelling.