This is a though one for me. I have lived here since 1982. I love Clarksville and can't imagine leaving it. Personally, I might consider Galatin. It looks like a great place to raise children and has all the conviences of Clarksville but it is 25-30 miles from Nashville. No much closer and traffic does suck up there. Personally, I think Springfield is a dump. Sorry, Margaret. But Pleasant Hill has some potential but I have never anything standout about the schools. I don't know your ties to Clarksville and I hate to admit it but Murfreesboro/Smyrna is somewhat comparable to Clarksville if the other side of Nashville is an option. I have traveled Middle Tennesse extensively so I have been all over. I don't sell homes so if you have a question about an area give me a call I will answer without trying to sell you a home.
PS. Disclaimer: I do provide mortgages. Being a related field it could be beneficial for me if you purchase a home. But that is not my intent if you ask a question about an area.
I have a new Neighborhoods section on my website that might help you sort through the areas of interest within Davidon county. Depending on budget, I like Donlelson for affordable & convenient family homes.