Do I use a website to download songs off of YouTube or Spotify?

Soulseek and/or private trackers, both with a VPN.

Where should I store the music?

On a hard disk?

I'm running GrapheneOS

Android isn't well suited for this. It's fine for playing either by transfering your files or streaming them with something like subsonic, mod, etc.

Maybe sync with SyncThing?

This could work yeah.

What's the best way to compress mp3 files but still retain the quality (even possible)?

By design MP3 is a lossy format, so no you cannot. What you want is FLAC, a lossless compressed format.