Hi,
I have lived in Springfield MO since November 2003. I was previously in Los Angeles for 35 years so my comments are "skewed". I fly out of SGF because its work related or using Frequent Flier miles. Some people I know drive to Kansas City, Tulsa or St. Louis for lower fares unless they fly Allegiant. There is a new ( May 09) airport
http://www.flybranson.com/
in Branson served by Airtran to/from Atlanta and Sun Country to/from Dallas and MSP.
The new Springfield Airport terminal also opened in May 09. It is very nice and modern but the old terminal was just fine in my opinion.
Regarding Branson, I spent 10 days there following an Ice Storm in January 2007 when power was out at my house in Springfield and we take out of town visitors there but its not really my cup of tea. It kind of reminds me of Gaitlinberg or Pigeon Forge TN. There is a country music focus but there are also some non-country shows. Since I don't go for country music I have omitted those but many people like that stuff.
I can recommend:
Shepard of the Hills outdoor pageant
Acrobats of China
Neil Goldman's Cirque ( Cirque Du Soleil knock off)
Dolly Parton's Dixie Stampede ( big production and I can overlook the music, many people actually like it)
Silver Dollar City Amusement park ( they have a good cave too)
Branson Scenic railway ( ask if they are going on the southern route, its preferable for scenery but they use working tracks so sometimes they go on a different route)
There are several factory outlet malls in Branson.
Of course you must visit museum with the Original Beverly Hillbilly's car:
http://www.roadsideamerica.com/story/14355
Table Rock Lake has the usual aquatic activities. There is a "River Boat" that goes on Dinner cruises and you can rent small boats.
Branson Landing is a new shopping area with 2 new Hilton Hotels. The Residence Inn is also nice. I also have to give thanks to the "Settle Inn" which allowed my 2 large dogs to stay there with us after the Ice Storm. It is a fine motel but more in a budget category.
On Highway 65 between Springfield and Branson you should stop at Lambert's Cafe for throwed rolls:
http://www.throwedrolls.com/shopcontent.asp?type=Ozark
Eureka Springs AR is a nice historic town and you should have a look at it.
http://www.eurekasprings.org/
Finally you mentioned:
fly to KC, then on to Branson or Springfield and rent a car.
There are no scheduled non-stop commercial flights from Kansas City to Springfield or Branson, so rent the car in KC.