Moving the camera while filming creates shaky images, but there are ways to cut down on this.
Getting Some Movement
It is true that it is much easier to keep the camera still at all times, especially since most independent filmmakers do not have the money to buy expensive additions like steady cam body mounts. Though any academic will likely tell you to stick with the stationary position almost exclusively, it is important to have the freedom to choose exactly how you want a scene to look. There are a number of ways to make the image remain stable while keeping the camera moving.
When you are trying to track something, such as a pair of characters walking or running, the best way to go about this is to walk backwards. Bring the camera into your chest and then walk slowly backward, placing your foot down on the ball of your feet and then the heel. Walking backward helps cut down on camera shaking and then allows you to get a useable image.
If you intend on moving the camera quite a bit you are always going to get a better video image if you use something to prop the camera. Anything with wheels and a flat surface will work, especially dollies or wagons. This will allow you to keep the camera strapped into the tripod, which makes things easier to work with when it is set onto the moving cart. Cars also work well, but really only when you are not going to use the natural sound and you want to get quick motion.
There are a number of software options to steady your video once you have already captured it. Occasionally these tools are in your editing package, but not always. Try to find a quality program online, especially since most are free.
The fact is that no matter what you do, any sort of camera movement will result in a certain amount of shake. This is all part of retaining an extensive amount of camera tracking. Without this it would feel less than spontaneous, and without expensive camera peripherals you are ensured to get some of this. Try out different techniques and see if you can reduce it as much as possible.