Going to be a very nice trip. Mount Rushmore, Yellowstone, then Oregon Willamette gorge to Portland. To get back to Taiwan, you may need to get down to San Francisco to get direct flight from China Airline or Eva.
RV will be pricey for sure, but worth the trip.
Or you can rent a sedan. Buy a 4 person tent. Stay in KOA along the route. http://koa.com/