I live in the area. I have to say that the Ancient Messene Archeological Site is the equivalent of what you would like to visit and see in Ancient Sparta. The monuments from Spartans have been destroyed or stolen during the centuries. Now there's almost nothing left in Sparta. On the other hand, the Messinians have not been "famous" like Spartans, so there is much to see. The poor people of Messinia had to endure the constant wars with Spartans. So, instead of visiting Sparta you can visit the Archeological site of Ancient Messene and learn both for Spartans and Messinians. The Archeological Site has many ancient buildings (theaters, stadium, etc). The site is a 20-30 minutes drive from the small Kalamata International Airport or 40 minutes drive from Kalamata.

Tips: 

Rent a car. Access by public transportation (1 bus) is not recommended and not convenient.

Visit the Arcadian Gate about 1-2 kilometers away. Do not forget to continue driving about 1 km and see the remains of the walls and the towers!

Always wear hat, sunscreen and have a bottle of water during the summer months. Avoid the hours 11am-17pm for a visit. Go earlier or later.

Wear comfortable shoes. No high-heels in Greek archeological sites.

During the summer months, some performances or concerts are organized in the theaters. It is a unique opportunity to combine a visit to the archeological site in the afternoon and stay there and watch the performance during the night and return back to your hotel in Kalamata or elsewhere after the performance. Most events in the area start at 8pm or 9 pm in the summer.

Best period of time: April to see the fields and the site being full of wild flowers. March may have some sunny pleasant days or rain and storms. The same applies for September. In July and August, do not visit the place after 10am, it's too hot.

Refill your bottle and drink water from the spring just above the archeological site and take a picture of the view above, at the road just above the archeological site.

A new museum is under development.