We go down there every year and just got back last week.
Yes, there is a lot of fun things to do -- my BIGGEST piece of advice is to let the kids know upfront that you will be doing xx number of paid things. Their eyes are going to be popping out when you drive down the main drag, there is mini-golf, go carts, all those funky museums, bungee jumping, etc, etc. Our first trip was filled with too many "no, we can't do that". After that trip the girls were told, "your choice of 2 items".
That said - here are my budget tips!
1) yes, yes on the National Park. Go to Sugarlands Visitor Center and see the free film about the Smokys. Sign the kids up for the Junior Ranger program. Check out the times of the free Ranger talks and walks (on the web site and at the Visitor Center) They have many hands on activities for the kids. Our favorite hike is the Grotto Falls hike in the Roaring Fork area. About 2.2 miles round trip to a waterfall that the kids can walk behind. Also, hit Clingman's Dome the highest lookout.
2) Tubing -- a day of fun for less than $8 per person. As someone else mentioned we used River Rage (they advertise $8 a day - green tubes). The other companies are about $13 and it is the same river. Yes, Rage had lots more people in the parking lot, etc. but they were very well organized and everything moved quickly. They have three drop off points - 1 1/2 hour lower river, 3 hour lower river, and 1 1/2 upper river. The lower river was less rapidy and less rocky. We did the 1 1/2 lower, had a picnic lunch by the river at the Rage place and then did the upper river. Bring water shoes for the rocks. Even if you don't do tubing you will want water shoes for the river fun in the parks.
3) Food - bring a soft cooler and pack picnic lunches. Make a big bag of GORP for the hikes and give everybody a baggie. Some places are just expensive. One day we felt like ice cream so we headed to Baskin Robbins -- it was $5.39 for a small milk shake

We couldn't do it, so we headed across the street to Fudruckers and got shakes for $3.79. You just have to be careful.
4) This was our first year doing a show, and we did Magic Beyond Belief. It was awesome and, like the Comedy Barn, kids were free.
5) Coupons, Coupons, Coupons. If you know before what you want to do go to their website and print a coupon. The River Rage site has coupons for $1 off -- so your day of renting is $7. Our favorite restaurant - the AppleBarn (a country restaurant with down home good food) has a loyalty program on line which gave us a buy-one-get-one-free lunch. When you get there you will see coupon books all over -- pretty much you should be able to find coupons for almost all attractions.
Too be honest, for us it's not all about saving the money -- we have more fun doing the items listed above. The area is beautiful and it's great to enjoy it. Have a great trip!