Welcome from a fellow gun owner in NC, just please don't bring the NJ politics or weather.
I live in the RDU area. I'm assuming you're moving into Wake county, but if you're not be aware that our sister counties(Durham and Orange County) don't have quite the same attitude towards guns as the rest of the state. Durham county actually requires registration of handguns, although I know more than a couple of people who ignore it.
Anyways -- as others have stated, go down to the sheriff's office, and you can apply for pistol permits and a CHP(must have a training course first).
we've got a nice indoor range that the county parks runs.
http://www.wakegov.com/firingrange/default.htm
here are two of my favorite stores -- good folks and fair prices
http://www.pdhsc.com/ (they also have a small range, pistol and rimfire only)
http://www.lawmens.com/