My question for you is: Why do you want to be rooted? Rooting does have benefits, but with the newer versions of Android, some of those benefits have been nullified. Like for instance, disabling apps. Now that can be done on stock Android through the Application Manager. Some you can only disable, not delete, but in the long run, it's better to disable/freeze before deleting anyway.
Rooting grants you use of some apps that require root access. Like Titanium Backup. It also makes putting a custom recovery on your phone a little easier. Custom recoveries ad depth to what you can do, like Paul said about wiping. Sometimes if a phone is running poorly, wiping the phone's cache and Dalvik cache often help long before a full wipe will. I am not totally sure, but I don't think full wipes/factory resets always get the Dalvik cache.
ROMs....bug thing here is make sure the ROM is for the Sprint Galaxy S3 (SPH-L710) Any other ROM put on your phone will most likely brick it. As for installing ROMs, most forums have instructions. If they do read them. When you are done reading, read them again. And once more time afterwards. Missed steps can lead to bad flashes, even possibly bricked phones.
Now, last piece of advice. GO to YouTube, and look up QBKing77. He does a lot of videos with the Galaxy S2 Epic 4G Touch, Galaxy S3, some on the Note 2, and some on the Galaxy Nexus. His videos are concise, full of information, links to files, related links to related videos, troubleshooting, you name it he probably made a video for it. I have found his help invaluable.
Sorry this was a little long winded. Hope it helps.