Messages are stored in the cloud. You don't have to do anything special other than log in on a computer and use Messages to get them. Are you saying you want to see your messages on the 10.6.8 machine?
Syncing synchronizes all items (contacts, calendar, music, apps, etc.) between the two devices. Backup creates a full backup that can be used to restore a device if you need to erase it and start over. Again, messages are stored in the cloud. I'm not sure if messages are part of a backup.
Just enable iCloud backup in your phone's iCloud settings and on your computer make sure iTunes is logged into the same account.
Steve