There are a few techniques to make things appear to be flying. I’m no expert but I like to learn about this things and I guess I have mastered one of them.
First of all, for HP lovers, I would suggest you, to get a wand, point at the object and after a flourish movement repeat, Wingardium Leviosa!
If this doesn’t work there are still other techniques to explore.
1) In settings where you can handle the lighting as much as the camera, the easiest way is to use very thin and transparent wire from which to hang the objects. The proper use of light and camera angle will make the wire invisible to the camera.
2) In these type of settings and also in those settings where the light can not be handled (like exteriors) there is another technique that I like to call the perspective technique. It consists in using hard wire or a stick or any thin but strong element that can support the weight of the flying object. This element must be placed in a way and angle that keeps itself hidden behind the flying object .
For example in picture 12 in this thread http://www.playmofriends.com/forum/index.php?topic=9117.0 there could be an element attached to one of the wheels of the plane and to the tree. This “element” would be hidden from view by the body of the plane. It is a difficult technique and it can’t be use with very small or very big objects but it is quite effective.
3) The best technique however for film and photographs is the use of an editing program, like Photoshop. It is not easy to explain how it works, but once you learn it, it is quite simple. I just use the CLONE tool. Basically this tool duplicates sectors of the existing photograph on the area of the photograph you choose. For example to cover my shadow on the grass I have duplicated over the shadow, sectors of sunlighted grass. And so on with the rest.
Here are some examples than are very explanatory. I hope this could help you.