One of our favorite places, especially if it's a rainy day...
www.monkeystreehouse.com My friend Natalie actually came up with this entire concept and it is a FABULOUS play area for kids!! She keeps it very clean too.
Of course the zoo is fabulous
www.nashvillezoo.org
I would think Adventure Science Center might be a little too much for a 4yo, but there is a cute toddler play area. Not really worth the price of admission, though, just to do that. The other thing I HATE about that place is that it is so hard to keep an eye on your kid when they enter the climbing structure up the center of the building. Always makes me a nervous wreck.
My kids LOVE to go down to the Farmer's Market and look at all the vegetables. There is a large state park outside
(http://www.tennessee.gov/environment/parks/Bicentennial/) where your son can run around, and a great food court with many different ethnic varieties of food!
http://www.nashvillefarmersmarket.org/
I agree with the PP about Nashville Parent. It's a great resource! There are free copies all around town, or you can access it online:
http://www.parentworld.com