If I download/save an image on iOS, how can I determine what the resolution of that image is?
There's no native way to do this within iOS without either installing a 3rd party app or jailbreaking the device.
The closest you can get natively is to determine the image file size (i.e. in KB or MB), but obviously this does not provide the resolution.
Another option is to import the images to your Mac/PC in which case you can use macOS, Windows, etc to display the resolution.
In terms of 3rd party apps, there are countless options on the iOS App Store that can do this for you. If you're interested in more than just resolution than you may want to download an EXIF viewer app, most of which are free.. These will offer a lot of other data from the image and often work with images that are downloaded, not just ones taken with your iPhone camera (although in many of those cases you'll only get a resolution, which is what you're after anyway).
There are also photography apps and file management apps that can give you this info, but the built-in iOS Files app won't provide resolution details, just the image type, file size and date.