Ok, I have my MP3s, but I also have some music I got off Amazon and Google Play alike. Can I just download it all to my phone/tablet/etc, and one player to play them all?
Eventually, yes.
The trick is that most of the streaming or download apps include enough DRM stuff in the files that you can only play the files in the player that you purchased or downloaded them in.
In some cases, you can download the files to your computer, which seems to remove the DRM bits, then copy those mp3 files to your device.
All my Amazon purchased and downloaded music is available through all players including play music. Music downloaded via streaming app only available through that app.