If sync is on, most everything should be automatic. Got into settings, find sync, and make sure everything that you want is backed up. Also go to your settings in contacts and sync to your Google account. If you go to contacts.google.com and all of your contacts are there, then you're good to go.
Google contacts are synced to the Google server, and any phone connected with your Google account will sync them. For other situations, see Backing up an Android Device - Back up the Stardust, restore to the On5. (The Samsung setup process on a new phone should offer to transfer a lot by using NFC - the "put the phones' backs together" thing - or see https://www.recovery-android.com/introduction-android-beam.html.)