If you go to the JC Parks and Rec page:
City of Johnson City, TN - Home
Click on the Fun Guide on the right side of the page. It opens in a new window. If you go to page 31, it is the start of the activities offered at the Princeton Arts Center. I looked through it quickly and I saw several activities for 3 year old children.
Also, I took my boys (age 2 and 7) to Into the Fire in Johnson City (in a strip mall on Oakland) and let them paint Christmas ornaments. They really enjoyed that. It was the first time my 2 year old had done it.
I haven't tried any of the classes at Princeton Arts Center. We just moved here last year and the after school ones start at 3:45pm. His school isn't out until around 3:30pm, so we could never make it there in time. I hope you find something you like.
I wish I knew where to get the Fun Guide in actual printed form!