As a very brief back ground, I came across the flat earth society by way of family member, and watched some 7 or 8 part video series to try and prove this theory. Personally, while I also have a high distrust in government, I'm remain totally convinced that the earth is round, and that not everything we were taught is false, particular the hard sciences, such as math, physics, chemistry etc.
On to the experiment. This is simple to do, one of my best friends in high school did this with his son for fun. Go out and buy a weather balloon, attach a video camera to it, and send it up. I saw the video my friend did, and the curvature is plain as day in it. He used an iphone 4, and ran that app to find your phone so they could locate it afterwards.