At the decree of King Otto of Greece, the present day city of Sparta was built in 1834 near the site of the original city. With its wide, tree lined boulevards, the city was designed to be a paragon for all Greek towns.
The modern town still reflects this original vision, and is a very pleasant and green city set before a backdrop of lush hills.
Bus: Buses between Sparti and Athens run a few times per day (see website of KTEL for exact schedule), and typically it takes 2 hours from Athens to Tripoli (a transport hub in central Peloponnese) and an additional hour onward to Sparti (in total 3 hours). The bus starts from the Kifissos bus station in Athens and arrives at the main bus station in Sparti, at the east side of town and around 15 minutes walk to downtown. The bus costs around 23 euros.
The city is quite small, and usually exploring by foot is the best option. From the city center all destinations in the city could be reached within 15 minutes walking.
To reach Mystra, take the bus that originates from a kiosk very close to city center (ask your hotel about its exact location and schedule, the hotel staff seems to have very good knowledge of it). The bus will drop you off in a parking lot at the foot of Mystra, and you will also wait for a return bus in the same parking lot.