How can I transfer MyBoy game saves from one phone to another?

You can through various methods:

Assuming you haven't altered the contents of your devices internal storage, the saves can be located from these paths:

sdcard/MyBoy/save/%ROMNAME%.sav

mnt/sdcard/MyBoy/save/%ROMNAME%.sav

storage/emulated/0/MyBoy/save/%ROMNAME%.sav

Note: "sdcard" and "0" both represent the storage root. Legacy devices typically use the mnt path.

All you will have to do is either copy your saves to a microSD card, or send them wirelessly through bluetooth or NFC. To simplify the process I recommend using any of the numerous free archiving apps like ZArchiver to compress your saves before transferring them. I would recommend using .zip format. You can easily accomplish this by long-pressing the save folder in ZArchiver and tapping Compress to *.zip. ZIP format is one of the most universal formats, and I haven't owned an Android smartphone that couldn't either create and/or extract the format.

In MyBoy settings, you can configure it to automatically sync your saves with a directory on your Google Drive account (in Settings->Misc->Sync to Google Drive / Sync Settings)

Well if you're using an android phone I'm pretty sure your saves will be synced via google play. Have you used a google id to log in? If so, use the same google id to log in via your new phone and the saved slots will be synced (This is assuming the games you play are downloaded and installed via google play).

The other way to transfer saved slots are to use a backup up like titanium or helium.