Transfer everything to new phone? 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:


Bluetooth: Bluetooth File Transfer
FTP: FTP Server, FTP Client
SAMBA: SAMBA Server, SAMBA Client


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.