I have been going there for 40 years. If you want to camp then Table Rock State Park is where you want to go. It is a true campground about 6 miles out of town and right on Table Rock Lake away from the hustle and bustle. If you want to stay in your RV and see Branson then I would stay at Branson Lakeside RV park which is owned by the city (make a reservation for a waterfront site, 100 thru 109). It is a park with full hookups. The sites are close together but it is right on Lake Taneycomo, which is really just a big coldwater river. The water comes from the bottom of Table Rock Lake and is 55 degrees year round. They keep it stocked with Trout and you can fish 30 feet from your RV. Branson Landing is a big outside mall right beside the park. They have a free shuttle service that will pick you up right at your campsite in a golf cart and take you to the Landing to shop or eat. All you have to do is tip the driver. There are lots of things to do in Branson. It is known for its shows (probably close to 100 different shows) and Silver Dollar City (a theme park). Silver Dollar City is a don't miss if you are there during the Christmas Season.
In the past I have stayed at most of the other RV parks and they all are good quality parks but our favorite is Branson Lakeside RV.
Shoot me a PM if you have any specific questions and I will be happy to answer. I can dig out some pictures of the park if you would like to see for yourself.
As for Eureka Springs, there is not much going on there these days. The old downtown in fun to visit but I would recommend a day trip down there from Branson. I think you would see all you wanted of ES in one day. It is about a hour drive from Branson.