Can I transfer all of my music, photos, apps, contacts and phone number from my old republic wireless moto x to a new republic wireless moto g?
In short, yes. Any user content synced to your Google account will sync back to your new phone when you add the same Google account used on your current phone to the new phone.
On your current phone, at Settings -> Backup & reset, please verify both Backup account and Automatic restore are toggled on.
For that which is not synced to your Google account, please try one of the following options:
Easy Backup & Restore: