From London, Paris, and many cities in England : Bordeaux (around 2h 30 by car to the Chateau). Bordeaux has the most direct flights. To shorten your drive, Brive la Gaillarde ( around 1h to the Chateau by car) or Bergerac (around 1h 15 to the Chateau by car)
From many cities in the US : Paris Orly Airport, then Paris Orly Airport to Bordeaux (then 2 hr 30 min drive to the Chateau).
Driving from Paris is approximately a 6 hour drive to the Chateau.
From Paris city center : Bordeaux then rent a car as there are no trains between Bordeaux and Sarlat until September 2019 because of renovations (around 2h 30 by car) or Souillac (45min by car)
From Paris Roissy CDG airport : Train to Bordeaux (3h 30) then rent a car
From Paris Orly airport : get a taxi to Massy Palaiseau TGV and the train to Bordeaux then rent a car.