Depending on your age, you will have to have a hunters ed course to get a hunting license. And if you plan to hunt, you will need at least a small game license, even with a pellet rifle.
http://mdc.mo.gov/hunting-trapping/hunter-education-and-safety/hunter-education-requirement
As far as the handgun goes, as long as you are 21, you can openly carry a loaded handgun in a holster on Federal Land and all of Missouri with the exception of a few cities that have an ordinace against it.
You would be perfectly legal, if you are of age, not a felon, and not under the influence of any drug or alcohol, to carry the firearm in a locked dry box or dry bag if you desire.
That being said, there are restrictions regarding the discharge of any firearm or hunting within in range of any campground, access, or boat ramp. I think there is one on there regarding the discharge of a firearm on the river also. If you are going to hunt, you will have to choose a place other than the above.
http://www.fs.usda.gov/recarea/mtnf/recarea/?recid=21676