For what it's worth, Google Play Music should work just fine for you on the phone. I haven't used it much, but I think it will pick up music and playlists from the default locations that MusicBee syncs to.
The important part for you is the page on devices/sync settings that frankz linked. That's where you will tell MusicBee which playlists to send to the phone and in what format (I suggest sticking with m3u, it's fairly universal).
Other than choosing playlists, I doubt you will need to change many of the settings. Maybe tell it to delete files not in the sync list? I'm not sure if that's automatically ticked.
To get into the settings while the phone is connected, you might have to open the left navigator, which i believe is hidden in the initial settings. You can hover over the left side of the window to get it to pop up. The phone should be near the bottom, you might have to scroll down.
I believe it also needs to be unlocked when you connect it - if it's not showing up, make sure that you have sync MTP turned on in Device Preferences (the preferences you can open from the main menu), unlock the phone, and restart MusicBee.
Once you've got the device settings in front of you (device info will be shown in the top half of the main panel, and settings in the bottom), go to the Music section of the settings. Untick Synchronize music, tick Synchronize playlists, and then "selected playlists only". Then select the playlists you want to sync. At that point you should be ready to give it a shot. You can always do a preview of the sync to see if it looks right.