There is much customization that can be done in Android without rooting (more or less the Android equivalent of jailbreaking). I don't know of a way to change the carrier name without rooting, however. Someone else may know of a way.
What kinds of customizations do you want to do? If you can give us some specifics, we might be able to point you in the right direction.
If you decide you need to root to do what you want, or just want to root for other reasons, there is something very, very important to keep in mind. Not all Galaxy S3s are the same. The international version is very different than North American versions, and different carrier versions are all different. Using a root procedure designed for one version of the phone on a different version can cause serious problems, including the potential of bricking your phone, which turns it into a very expensive door stop. Flashing an international ROM on a NA phone, or vice-versa, is guaranteed to hard brick your device, and flashing the wrong carrier version can do so.
If you decide you want to root, go to the rooting subforum for your specific device, and read and ask for advice there. If you find instructions or ROMs on other sites, or on Youtube, don't even think about using them unless you're 100% positive they are intended for you exact model and carrier version of the S3. We see way too many bricked phones because people just don't pay attention to what they're doing.