In TN, gun carry of any kind is illegal without a permit. If TN recognizes your states permit, then you are good to go.... With a permit you can carry a handgun, open or concealed. Open carry or concealed carry of long guns is prohibited though, its only for handguns.
As of July 1st its legal to carry any firearm you want inside your own car without a permit... as long as you're legally allowed to own the gun and drive that vehicle (unless its a company owned vehicle and the company has a no-gun policy).
With a permit, carry is allowed in state parks and civic centers etc... except that whoever runs the park is legally allowed to override that state law and post the park as no carry... so if your going into a park you need to check for signage and/or call and try to find out if carry is not permitted in the specific park you are going to. There are other summaries of places you cannot carry at handgunlaw.us
As to "no gun" signs... if a sign in text specifically mentions the law and where it can be found in the law that they are posting no guns... or if its some type of traditional gunbuster sign (any color, any size, any look, as long as it is a circle with a slash and somewhere inside that circle has an image of some type of firearm), then it is against the law to carry into the posted place. I believe its a class b misdemeanor if you are caught and charged... if I recall correctly.