Sparta is best reached by car or by KTEL bus. There is no train; the nearest airport is Kalamata (about ninety minutes by road over the Taygetos), and Athens international airport is roughly three and a half hours by car.
By bus, KTEL Lakonias runs frequent services from Athens (Kifissos terminal) — about four hours, several times a day. From Kalamata, services are less frequent but reliable. Within Sparta, almost everything is walkable; for Mistra, hourly buses run from the central square — a fifteen-minute ride — or a taxi is around 10 euros.
If you have a car, the drive south from Athens via the Korinth canal, Tripoli, and the new motorway is straightforward and about three hours door-to-door.