Upgrading from an older, slower, not nearly as good-looking Android device to a newer, faster, better-looking Moto X isn't a bad decision. Often the only drawback to changing devices is the process of transferring important information such as text messages and call logs to the new device.
To begin migrating your data to the Moto X, you'll first need to install Motorola's Migrate on your old device. It's available to download in the Play store. Droid users can also take advantage of this same feature, as by Dan Graziano.
Once you have Migrate installed, launch the Migrate app on your Moto X and pair the two devices. You'll do this by scanning a QR code on your Moto X with your old phone. Both devices will have to be on the same Wi-Fi network in order to communicate.
After the two devices are paired, the sync process will begin. You can expect your text messages, call history, SIM contacts, Media, and finally the volume & brightness settings to be transferred.
The time it takes to transfer your data depends on the amount of data you have on your old phone. You'll be able to use both devices while the transfer takes place. There will be a progress indicator in the notification tray to keep you up-to-date.
After the transfer finishes, you'll find your information on the Moto X just as it was on your old phone.