My wife's iPhone 5S doesn't have enough storage space for all of her music. If we were to buy a USB drive that connects to her phone, could we put her music on that and then have the phone play it from there in a reasonably hassle-free way? What would the process be?
You can read a flash from an iOS device, but this capability is limited to some very fancy lightning-port enabled USB flash drives.
For most use cases, I would recommend using a cloud service like Google Drive if you're looking to listen to purchased music on an iPhone without using up storage space. Essentially, you can just drag your music purchased from anywhere into Google Drive, download the app onto your iPhone, sign into your account, and listen to your music (though you of course need an internet connection unless you download them to your phone).
However, if you want to buy a special flash drive, you can purchase the Sandisk iXpand Flash Drive, which plugs directly into the lightning port and can play music while plugged into the iPhone, through the Sandisk iXpand app. Just connect it to a Mac/PC beforehand to load music onto it through the standard USB port.
Edit: don't forget about streaming services like Spotify and Apple Music, which don't take up storage space either.
I don't use a USB drive but do this with a My Passport Wireless Pro drive. The iPhone connects to the My Passport through the Passport's Wi-Fi and the My Cloud app gives me access to all the music and audio books we own. The only issue I have is that I can't give the media player commands through Siri. Honestly if Siri doesn't start to learn the way Alexa does my next phone won't be an Apple.
If I didn't use the My Passport for other things it wouldn't be worth it, but at this point I still have a need for external storage.