are there picnic tables here?

Yes

At the ranger station yes

There is a visitors center at one of the picnic areas that is great for kids. A playground as well.

Yes. The park is located in 4 different counties covering over 25,000 acres. Picnic tables are located at the visitor center in Monteagle. There are 8 other small picnic areas with grills located throughout the park. There are 4 picnic pavilions for larger parties that can be reserved in advance.