Try Virginia/North Carolina
Euan, If you want to spend 2 weeks cycling over here, I'd strongly suggest riding in Virginia and North Carolina. There's the Blue Ridge Parkway, one of the most beautiful roads in the world, (469 miles in the mountains) to some beautiful beaches along NC's Outer Banks. You could start in Northern Virginia/Washington D.C. and see the history of the USA. There's a lot of history in these 2 states, including a large Scotch-Irish presence in the Virginia and NC mountains.
I've never found such diversity of terrain and friendlier people than in these two states. If you like hills, I'd go with the Parkway. If not, try Norfolk and The Outer Banks along the coast. There are beautiful lighthouses, golf courses, small towns, big cities, and great B&B's all over the states. Use your local search engine for "Blue Ridge Parkway" and "Outer Banks" and do some research. Good luck...