I just redeemed my Apple Music, but my purchased music isn't in my library. If I try to download them again from iTunes, it still won't show up in my library.

By "redeemed" do you mean you started your Apple Music subscription or renewed it as it was set up before? Never heard any refer to Apple Music with the word "redeemed" before.

Either way, once you have Apple Music turned ON, any music you purchased through iTunes should automatically show up in your Library - assuming that you don't have "Only Downloaded Music" turned ON. If you do, then try turning it off.

Mobius One wrote:

hmm... I renewed it.

Where's "show downloaded music?"

oh, one thing: when I go to the "purchased" section in iTunes, no matter how many times I redownload the song, it's still stuck in the "not in my iPhone" section.

In the MUSIC app - not iTunes - tap on My Music at the bottom. Then tap on Library at the top. Towards the top (just below the Album or Playlist icons) there is in red type the view you have selected (might be Albums, Artists, etc.). Tap that and at the bottom of the drop down list it says "Only Downloaded Music". Make sure that is turned OFF to show everything you have in your Apple Music/iCloud Music Library.

Not sure about what you are seeing in the Purchased section of iTunes as I never have downloaded music directly to my iPhone from iTunes - I do it through iTunes on my Mac. But assuming that your iTunes account and your Music App are signed into the same Apple ID, any songs you purchased through iTunes should automatically show up in your Library in the Music App - again assuming you make sure that Only Downloaded Music is turned off.