Greenville-Spartanburg (GSP) is the closest airport and a 20 minute drive to Greenville. You can get an Uber/Lyft, taxi, or rent a car from the airport.
*If you are flying from the DC area - Washington Reagan National Airport (DCA) and Baltimore Washington International (BWI) both have non-stop flights with American and Southwest.
From Dulles (IAD) they have flights with 1 stop on American, Delta, and United.
Charlotte Douglas (CLT) airport is about a 1.5 hour drive to Greenville, SC. You can rent a car at the airport and drive to Greenville.
Brandon and I have done it many times and it's a pretty easy (and pretty!) drive down. You start on 29 South and go through Charlottesville (if you want really good bagels, I suggest a stop at Bodo Bagels - Emmet St. one) and then continue your journey to 85 via North Carolina and finally you will have arrived in Greenville, South Carolina.