If you are not rooted and do not want to root you have a few options for a partial backup. To start you will not be able to backup your saves (for example the finished levels in angry birds or levels in "cut the rope") without root. So if that is your goal, you need to root your phone, but then the warranty is voided, so
we will not go there
.
To backup your text messages, download the app
SMS Backup by "Michele Valanyini" it will backup your txt messages to a file on your internal memory card so you can restore it later. Then download the app
App List Backup this app will create a list of all of the apps installed on your phone.
Save all MMS pictures to your memory card. Once that is saved, you need to backup the stuff on your internal memory card, as that cannot be removed and moved to the new phone.
Now once you have the new phone, copy the stuff from your internal sdcard to the new phone, install
App List Backup, run it and it should find your old list so you can reinstall your apps again.