I have used MyBackup Pro for several years on different phones. It costs a few bucks. I've had to restore to prior phones, but not yet to the Moto X. It will backup and restore text messages, contacts, and apps (the restore process on apps is a bit tedious, you have to confirm many of them one at a time). It backs up music and photos as well, but I usually store those on an external micro SD card and only backup the phone's internal memory using MBP. All my music is on my computer anyway, and I'll backup photos separately.
MBP does not backup and restore your log-in information for email, dropbox, and other accounts. It also does not restore home screen layouts - I am now using Nova Launcher, which I like a lot, and I think it backs up and can restore your home screen layouts of apps & widgets.
If there's a more comprehensive backup option (that does not require rooting), I'd like to hear about it.