Why exactly do you want to remove them? If you are worried about RAM/CPU or just don't want to see them, it's a lot easier to just disable them in the application manager. Menu -> Settings -> Application Manager -> All. Find what you want to disable and click it. Select Disable. ( If you have installed updates, you must first "Uninstall Updates", found in the same exact spot. )
If you still wish to fully remove them to get back the 100 or so megs they take up, you will want to look around in places dedicated to root. On these forums, you can see it right at the top. You mentioned being on Verizon, so you can find the root section here:
(Verizon) Galaxy S3 - All Things Root - Android Forums
There is also the more advanced xda forums.
Samsung Galaxy S III – xda-developers
I don't know enough to specifically help you out here. I believe you can simply remove the APK, but am not certain. There will be some ROMs that have the bloat removed already, but then you don't get to pick and choose what you do or do not want. Also, flashing a new ROM will wipe your phone back to factory. Depending on how much you have done, this can be a pain. However, there are apps to help you with this, such as Titanium Backup.
If you have any more root related questions, it is best to keep those to the above mentioned "All Things Root". This general section is for stock S3 questions. ( This includes anything though that you have questions about that is NOT root related, say battery life or changing your launcher, etc. You're not excluded from here now that you are rooted, hehe. )
David: Rooting does NOT necessarily remove anything. It is perfectly possible to root and keep the stock ROM. It is even possible to flash ROMs that still contain the bloat.
Hopefully that cleared some things up!