This has to be the absolute best internet radio app.
https://play.google.com/store/apps/details?id=com.audials
It's free, no ads, not even a paid version anymore! Over 250 thousand stations & podcasts. The app has a record function, fabulous sound with built-in player & equalizer. Tons of settings, search, etc. And did I mention it's blazing fast? TuneIn used to be my go to app, but haven't used it in many months, and after Play Music served me ads every other song the other say, I'm done with them too.
I do have a nice little internet app, open source, free, no ads, called "internet radio" I picked up on
F-Droid , and it might be in playstore too. But, no record function. Very light app though, and worked rather well.
Internet radio has come a long ways in the last few years. But, be ready to expend about 50-100 mb per hour. Or stream over Wi-Fi only.