First of all - if you haven't been to Arrington Vineyards, you NEED to! It's beautiful, close to the city (30 minutes from downtown) and the wine is amazing!
But if you're looking for a day-trip wine excursion, there are some other wineries within a few hours of Nashville!
Sevierville, TN (3 hours, 20 mins)
Hampshire, TN (1 hour, 10 mins)
(There's a tasting room for this winery at Fontanel if you don't feel like making a long trip!)
Clarksville, TN (50 mins)
Gatlinburg, TN (3 hours, 50 mins)
Pigeon Forge, TN (3 hours, 25 mins)