After decades of neglect from its forced closing by Soviet authorities, St. Nicholas Monestary is up and running again. This really is an off-the-beaten-path attraction -- it didn't seem to me that they really get any visitors at all. Sadly, many of the buildings are still pretty run-down, but they're in the process of renovating their main cathedral, so perhaps in a few years this monestary will be more a draw. But some of the church buildings have been at least partially restored and are open. You can buy sweet-smelling prayer candles made of honey and wax straight from the monks. There is a fresh water well here that some claim has holy healing powers, so be sure to bring an empty bottle with you if you want to take some home!
Ask around enough, and someone will take you to Novomoskovsk's Ukrainian saddle horse ranch, where horses are bred for international competitions. It's a pretty low-budget affair, but they're very proud of these uniquely Ukrainian horses, a breed only created after WWII. They're big horses, but friendly and tough, and for just a few cents they'll let you ride around in one of their practice rooms.
Novomoskovsk's mail claim to fame is this wooden cathedral, supposedly the largest wooden church in Europe. This massive 9-domed structure was made out of interlocking wooden pieces in the 18th century, and amazingly still stands firm today. The grounds could use some substantial restoration and modernization, but it's still an interesting place to visit, and we found the staff very friendly and welcoming.