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Any of the major chain hotels around the airport should be fine for families.
As for outdoor "must do's," I recommend the Nashville zoo, Percy Warner Park, Radnor Lake to name a few. The latter two have many miles of hiking trails through the woods. I've not been to the zoo, but I hear it is one of the best in the southeast. Also, the greenway system is great. There is a trailhead down the Stones River at Percy Priest Dam, which is near the airport. Other trailheads are scattered throughout the city as well.
Nashville in early April averages around 65 degrees. I've seen it anywhere from 30-80 at that time of year; but I'd put my money on it being near 65. So if that sounds comfortable to you, then sure.
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