For reference the zoom of the lens is simply the big number divided by the little number. so for an 18-55 lens it is 55 / 18 which is simply 3X zoom. Why the apparently low numbers? Because with SLR cameras you have a choice. If the current lens doesn't do it for you, then you change it! By making lenses with a smaller zoom, it's actually possible to make them better quality, because there is not so much compromise to work at a wide range of focal lengths. Having said that, you can buy 18-200mm lenses, so that is quite a large 11.1X zoom.
Moving into SLR photography you need to forget about this magic zoom factor and concentrate on focal lengths.
A focal length of 50mm is regarded as "Normal" and all other references are drawn from here. This is something supposedly similar to what the human eye sees. Normal focal lengths are used for portraits, although up to 85mm is still considered a portrait lens.
Any number lower than 50mm is considered a wide angle lens, and any number over 85mm is considered telephoto. SLR lenses can have fixed focal lengths, or they may zoom. An 18-55mm lens is wide angle to normal zoom lens. A 70-200mm lens is a zoom telephoto lens. Notice how a lens can be a zoom lens but not necessarily a telephoto lens, this is a popular misconception.
Your 55-200 mm lens is therefore basically a 4X zoom that starts at normal and goes into the telephoto range. That lens off the top of my head should be able to focus from about 4' up to infinity. The further away a moving object is fro you the more light will be required to get a non-blurry shot. For example at 55mm a slow moving object will produce a reasonable shot down to about 1/100th of a second. at 200mm a fast moving object may still show some blur at 1/800th of a second. In which case you will need at least 1/1000th of a second to get a clean shot.
My recommendation out of all this is to work around at least 1/800th of a second or faster by selecting Shutter Priority Mode (S) and then if you can't get 1/800th of a second move the ISO up until you can. Don't go beyond ISO 800 though. If you still can't get 1/800th of a second then I would think about moving closer or giving up.