You will have one severe grade known as Saluda Grade just after you get into NC. All uphill but shouldn't be a problem. I usually stay one lane from the right and manage 50-55 mph. If you do decide to get in the right lane, watch out for the semis which can get as slow as 15 mph so you catch them very quickly.
Next will be the dip into Green River which has a posted speed limit of 55 to help reduce accidents in the prestine watershed. You may wan to down shift one gear to let the engine help hold the speed down.
Rest of your trip will be pretty peaceful.
Biggest thing about mountain driving is to keep your speed down a little more, downshift as you crest the top of the big hills (usually you bleed speed at the top so you can downshift two gears), and don't ride the brakes. A good rule of thumb is you will want to use the same gear going down as you used climbing the hill. And remember, it's easier to upshift to reduce engine braking than it is to downshift when you find you need more.
But don't worry about this trip, while it is a good practice run to learn how to do some things, there really isn't any hills that will get you into trouble as long as you hold your speed down within the limits.
Have fun.