Me and the wife went just a few weeks ago. The Parthenon is kinda cool, but doesn't take to long to walk through. We took one of the tours of the city on a trolley. They'll let you off at any of the stops so you can look around and the trolleys run about every 20 min. We made a day of just hoping on and off the trolley and checking out various things. You can always go hang out at the BassPro up there.
White Castle. Theres a place outside of Nashville called Loveless Cafe, family style breakfast was NO JOKE! We just walked around the downtown area. We stayed at the Gaylord Opry which is huge and very nice.
My wife and I love Nashville, we go up once a year for a 3-4 day weekend. Things to do:
The Opryland complex is not near downtown, it's out in the suburbs. But, there's Opry Mills Mall (where Bass Pro Shops is), which is huge. I'm not a "shopping" type of guy, but it's a nice place. The Grand Ol Opry House is next door to the mall. They have shows on Tues, Friday, and Sat. The Tues shows usually have a big name performer (since weekends are when they play their touring shows). The weekend performers tend to be old timers and has been's. The Opry Land Hotel is here, as well. Even if you're not staying here, it's neat to look around. I should mention that everything at the complex, the mall, everything, is brand new and refurbished due to the flood a few years back. So, everything is nice. Pull up some Nashville/Opry flood stuff on YouTube and watch it with your kids, then check it out now.
Downtown is great. On Lower Broadway, you've got 7-9 honkeytonks, all with live music, all free of cover, and I know some of them are kid friendly until a certain time in the evening. There's the usual t shirt/boots/cowboy hat tourist stores. There's Margaritaville and a Hard Rock to eat at, and there's a place across the street from the Hard Rock, I forget the name of it, that us good. Also, there's Jack's BBQ. It's good, not great, but it's a neat hole in the wall type of place.
I highly recommend taking a tour of the Ryman, and pay a bit extra and do the backstage tour.
I also highly recommend doing the Country Music Hall of Fame. **Pay the extra and do the RCA Studio B tour** They're doing a massive expansion of the HOF, it's suppose to be done this month. Also, check out Hatch Show Print, it's in the same building as the HOF.
I also recommend riding out to the Loveless Cafe. You'll need a GPS to find it, it's off the beaten path, but it is slap yo momma good home cooking.
That's a few things, but maybe it'll give you some ideas.