Some things to make sure when flashing roms.

Make sure the rom you are trying to flash is for your device AND carrier (Verizon in this case)
Make sure the md5 of the download matches the md5 posted at the download site--see this about md5 checksums--http://androidforums.com/faqs/443074-md5-verification.html
More often than not, there will be flashing instructions at the rom website...but generally, this is my procedure:
Make sure you save your contacts to google. Save any apps/data etc to google or use Titanium Backup (an awesome app in the PlayStore)
Make sure rom is on the sd card --internal or external.
1. Make a nandroid backup.-- in recovery.
2. In recovery wipe cache/dalvik cache/factory reset/wipe data.
3. Flash rom.
4. Reboot.
Now, some roms such as AOSP, require a separate download of GAPPS. (google apps) after the initial rom flash. Carbon is one of them I think.
Edit: looking at your info, I would assume you have the Verizon S3 and not the HTC thunderbolt?
