The Asheville airport is about an hour away from Hot Springs and has flights to and from Charlotte, Atlanta, Pittsburgh, Raleigh and Cincinnati. Knoxville Airport is one and a half hours and has flights to NY, Washington, Chicago, Atlanta, Orlando, Dallas and a number of other cities. You may want to check both as costs for the same destination can vary significantly.
Driving
From 1-40, coming from the East
As you approach Asheville, take exit 53B onto 1-240 west. Go 5-6 miles on 1-240. Take exit 4A onto Highway 19/23 North toward Weaverville. Go 8-10 miles until you see a sign that says Hot Springs 2nd right. Under the underpass exit right onto Highway 25/70 going north toward Marshall. Follow Highway 25/70 north about 30 miles to Hot Springs.
From 1-26, coming from the South
1-26 will turn into 1-240 as you approach Asheville. Go 4-5 miles on 1-240 to the downtown area. Take exit 4A onto US Highway 19/23 North toward Weaverville. Go 8-10 miles until you see a sign that says Hot Springs 2nd right. Under the underpass exit right onto Highway 25/70 going north toward Marshall. Follow Highway 25/70 north about 30 miles to Hot Springs.
From 1-40, coming from the West
As you approach Newport Tennessee, take exit 432 onto highway 25/70 headed east. Follow 25/70 east through Newport about 25 miles to Hot Springs.
From Ohio and the Midwest
Most travelers from this area take 1-75 south. A family member who comes here regularly has found that exiting at Corbin Kentucky and taking 25E 116 miles to Newport is a good route. The exit number off of 75 is 29. 25E goes through Barbourville, Pineville, Middlesboro, Tazewell and number of small communities. In Tazewell there is one place where 25E turns left and if you miss the turn you can go way out of the way. When you arrive in Newport 25E becomes 25/70. Take 25/70 26 miles east to Hot Springs.