On android how can I have music player and Pokemon GO active at the same time?

I have been able to do this with several different apps that play sounds, and this process seems to work.

1. Start up your music player and start playing. (I don't know why you need to do this, and you might not have to, but in my experience it works better this way, so I do it.)
2. Use the home button to get back to your home screen, and launch PO:GO. It will eventually take control of the audio from your music player and the music will stop.

Once PO:GO is loaded and you're on the map screen, you can safely restart your music player. If you're lucky, as @Kevin notes, your player will insert playback controls into the tray, and you can use those. If not, you should do the following:

3. Pull up the task list (long-press on Home, or whatever the keypress is for your phone) and switch back to your music player. It should be stopped, just restart it.
4. Pull up the task list again and switch back to PO:GO. It will not try to recapture the audio again if it's already running.

The key is not to shut down either PO:GO or the music app once you have started them up; just task switch between them. PO:GO will give up the audio control when your music player asks for it, which it should do as soon as you tell it to start playing.