We recommend Greenville-Spartanburg International Airport (GSP), Asheville Regional Airport (AVL), or Charlotte Douglas International Airport (CLT). Charlotte has the most flight options, but is half an hour further. Other NC airports in Greensboro (GSO) and in the Triangle area (RDU) would be a longer drive, but may have good options depending on where you're coming from.
If you’re not renting a car or carpooling, there is a shuttle service called the Hickory Hop that runs from the Charlotte airport and could get you to Hickory (~30 minutes from Morganton) where you could grab a Lyft, Uber, or cab to Morganton.