Two ways to go: Train and bus, or all bus. Bus is cheaper. When you get in at the airport in Washington DC, ask the information desk how to get to the train station or bus station in DC, and you'll probably have to take a taxi to get there, or they may even tell you about a "shuttle" that will ride you over there.
If you take the train to the beach, when you get to the train station, arrange for a ticket that takes you to Norfolk VA. When you get to Norfolk, take a taxi to the bus station. At the bus, arrange for a ticket that will take you to Kitty Hawk NC, which will probably include a stop in Elizabeth City NC. They may tell you the bus only stops further up the beach from Kitty Hawk, like Nags Head or something. That's fine, just get to the beach. When you are let off at the beach, if you are far away from your hotel, you can always go to the nearest telephone and call your hotel, and they will help you get the rest of the way.
If you don't take the train to the beach, and you want to take the bus the whole way, when you get to the bus station in DC, arrange the whole trip from DC to Kitty Hawk NC, they will set up the times, cost, stops, everything, and they will give you a ticket for the whole distance.
There is no way to get to Kitty Hawk except by bus. Also, once the bus gets to the beach area, you sometimes can ask the driver to stop on the roadway and let you out wherever your hotel or cottage is, and he should be able to drop you within a few blocks of there. If you have a lot of luggage, well, might be better to get off at the "main" stop, and then telephone the hotel for help.
If it were me, I'd take that train to Norfolk. This is because bus rides are so tiring, a lot of stops and layovers, usually crowded. That way, you split up the trip a little bit. If you have a long layover in Norfolk before the bus will take you to the beach, you cannot just wander the streets of Norfolk at nighttime. It is just too dangerous, in ways you cannot begin to imagine. Even in the bus station, you must always be careful of your money, you must try to stay together, because bad people will sometimes hang out there and bother you. Be cordial to people in public places, but not overly friendly. No need to be afraid in the station; just be self-aware at all times, never let your guard down. But now, if you have a layover in Elizabeth City NC, it is much safer there, but don't go anywhere without first talking to the ticket agent about what you're doing. He can tell you if you're heading into any dangerous areas and so forth.
I don't know how solid your plans are for going to Kitty Hawk, but if you want to experience the beach, you might consider just taking the train from D.C. straight to Virginia Beach VA. That train station is just a couple blocks from the ocean! It's a nice, big beach, lots of places to stay, a boardwalk, and you have the advantage of being able to take a train the entire distance, and not have to fool around with switches to a bus, layovers, and all that stuff. Also, if you decide this is a good idea and you are going to stay awhile in Virginia Beach, you could always take a "day trip" or "overnight" down to Kitty Hawk, and your hotel consierge can make the roundtrip arrangements for you.
I hope this helps. There are websites devoted to "student travel," if you will Google around for that. It is good you are trying to prepare for this visit to the U.S. It is hard for people who have never been here to fully comprehend how large and complex this country is. Be sure to carry a very good map of the Eastern United States, become familiar with the cities you are going to be in, and pay very close attention to your money and your personal safety at all times. Then smile and enjoy!