I am changing from one Galaxy S3 to another S3. How can I transfer ALL my stuff? (I am not rooted.)
Your apps will be synchronized with your Google account. You can see what apps are synchronized by going to the web-based Google play/android market site and going into account settings.
Your contacts as well are synchronized by default but to make sure, go to account settings on your phone.
Your pictures, downloads, and any other files you may want to transfer can be uploaded to your computer temporarily and then from your computer to your new phone.
Alternatively, you could do this last bit a number of ways:
You get the idea: File Transfer Apps
You would need to root your phone to be able to access or copy the application data to your new phone.
There's an app that can do this, un-rooted, called DroidCopy
It transfers everything across via Bluetooth.
Disclosure: I am associated with the app.
Helium - App Sync and Backup (previously Carbon) will sync all your app data, such as settings and so on: