Flight
If you are planning on flying into the area the Durango airport is the best option for several airlines. Although, there are also flights to Cortez directly from Denver or Phoenix your selection and pricing will be better through Durango.
Hotel
Consider staying at the recently renovated Mesa Verde Motel located in Mancos, CO. It's a bit of a drive, around an hour, but it's situated in a great area. Alternatively, there are several AirBnB and Hotel options in Cortez, CO that are closer.