Summarize:

This is the gear-shaped icon that will be on one of your home screens or in the App Drawer.  This is in the orange section of options. Doing this will make sure your contacts are backed up to your Google account.    This is directly above Backup & reset in the orange section of options.         Your other Android should now be retrieving data from your Google account--including your contact information.
Open your Android's settings. Tap the Personal tab. Scroll down and tap Backup & Reset. Slide the Back up my data switch right to the "On" position. Unlock your other Android device. Open your other Android's settings. Tap the Personal tab. Scroll down and tap Accounts. Tap Add account. Select Google. Type in your email address. Tap Next. Type in your email's password. Tap Next. Tap Accept. Check the Automatically back up device data box. Tap Next.