C.H.
Hi T.! There are some playgroups in Clarksville. When I first started looking a few years ago, I didn't have much luck. However, I finally found MOMS Club of Clarksville. The link is http://www.geocities.com/clarksvillemomsclub/. The link will give you some info about the club. They try to have at least one activity a week and there are also smaller, age-specific playgroups as well.
There's also the library on Pageant Lane and Chick-Fil-A - just for general indoor play time for kids. I don't know how familiar you are with the area, but once the weather warms up, there are a lot of great parks in town also.
I don't know if you're military or not, but I believe Fort Campbell has some playgroup/mom focused things to do as well.
I hope you enjoy Clarksville! We've been here just over 4 years and we're leaving in June. This is a place I will truly miss. Best of luck to you!