Here's the strategy I (and probably others) would recommend.
For your first trip, book a charter to the INDRA, which is a little closer, shallower, and gives a nice introduction to the boat, the lines, and the diving off NC (along with some cool stuff to see). These are usually afternoon charters on weekend days, so you can spend the morning doing something with the family.
Then, book a full day charter and pick some kind of specialty class that you want to do along the way towards AOW (Nitrox is great if you don't have it already). The key here is that you should get to dive with your instructor as a buddy - a real plus for your first offshore dives.
If you have a choice, schedule the dives for early in the week. Then you can reschedule for later in the week if weather cancels the dive trip.
Olympus Dive Center (who probably has an ad on the top of this page) and
Discovery Diving Co., Inc. - North Carolina Wreck Diving in the Graveyard of the Atlantic are the two major dive ops in the region. They both do a nice job, IMO, and it's easy to find good reviews about both.
Have a great trip-
Rich