Apple Music Radio has eliminated much of what made iTunes Radio interesting. The composer-specific stations in the Classical genre contained well-selected music focused on individual composers. Many of the Top 50 stations also had good selections. And I used to listen to several of the jazz stations. You could fine-tune radio stations, adding and excluding certain songs or artists, and even share your stations.
All that is gone. Apple assumes that you don't want that type of listening, but would rather listen to "curated" playlists of music. That's fine if you want to listen to an hour's worth of music (see the next question below), but if you used to use iTunes Radio as your daily music source, you'll need to change your ways.
Q: I like Apple Music's For You recommendations, but I'd like them even more if there were some way I could click the Play button and listen to them all. As it is, I have to play each playlist or album on its own. Is there any way to just listen to the whole list? And, bonus points if there's a way to play this music in shuffle mode.
Yes, you've found one of the lacunae of Apple Music. While you can "discover" lots of music, it may not be presented in the way you want to play it. As you say, you have to start playing each playlist or album that Apple Music recommends.
There's a workaround, but it's not really much better. Start by creating a playlist; call it whatever you want, perhaps something like For Me.
Click each playlist or album in the For You section. Click the "..." button to the right of the playlist or album name; click Add To; and then choose your playlist.
This will take some time, but when you're done, you can go to that playlist in iTunes and play it either in order or in shuffle mode.
This also has the advantage of collecting all your recommendations in one place. You may want to add certain songs or albums to you music library, or you may want to rate some of them. You'll be able to come back to them later, after they've been superseded by newer recommendations in For You.
Note that you'll need to turn on the iCloud Music Library to be able to add tracks from Apple Music to your library, or to a playlist. However, you might not want to do this, because iCloud Music Library can wreak havoc on your music library.
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