Having trouble getting action shots with my camera?

I have a Nikon d200 with an 18-55mm lens. I bought it to take photos at my job at birthday parties for little kids. Only problem is, my very first party (it was only for practice and to get some photos on the website) I don't have any usable photos from when they were doing a lot of moving around for dancing and games (which is almost the entire party). Now, the lighting was really bad and dark but I can't just not take photos because it is too dark if someone pays for it. How can I capture the movement better? I had my camera on both manual and shutter priority but with both because it was so dark, I couldn't raise my shutter speed enough and still get a well lit, blur-free photo. Thank you for any suggestions!

Added (1). I DO KNOW what I'm doing. I'm not completely clueless about photography. This is a new camera I've never used before and I don't have the money to go out and buy a bunch more equipment for it. I need to work with what I have.

If your shutter speed is low during fast movement you may as well accept that your photo is going to be blurred. In order to get a good clear action shot in a lower light environment you need to create artificial light. Invest in a really good flash or possible external lights,

Sorry it is not more simple then that

Your shutter speed is too low.

Either use a flash or get a faster lens that will allow you to shoot at f/1.4. The 18-55 kit lens is too slow.

Usually for low light I tend to shoot aperture priority mode and let it auto set the shutter, I open up the aperture as much as possible. I see what it wants for a shutter speed if it's anything longer that 1/30th or so I try and raise the ISO, given the shot's will be grainier, but they won't be blurry.

As you've quickly discovered having the right equipment for the job is important and at this point in time you really don't have the right equipment. An 18-55mm f3.5-5.6 zoom just won't get the job done indoors without using flash and the little pop-up flash will not do the job.

The easiest solution is to buy a flash and learn how to use it in bounce mode. The Nikon SB-600 is an excellent choice. Don't even think about the SB-400 because it doesn't have a tilt head and you need the tilt head for bounce flash in horizontal orientation. This article will help you learn to use bounce flash:
http://www.wikihow.com/Use-Bounce-Flash-to-Improve-Your-Photography

Here is a video on using the SB-600:

When using bounce flash isn't possible you'll need a flash diffuser. This article will help:
http://www.digital-photography-school.com/using-flash-diffusers-and-reflectors

Another option is to buy the Nikkor 50mm f1.4 lens and use an editing program to deal with any excessive digital noise. This article will help with noise:
http://www.shutterbug.net/equipmentreviews/software_computers/0309noise/index.html

If most of your photography is going to be indoors in people's homes you might want to consider the Nikon 24mm f1.4 lens since a 50mm lens is a little long for the average living room/family room.

As you can easily see, you're going to need to invest some money if you expect to be successful with your photography business.

If you are going to offer a photo service for payment then for goodness sake get your technical act together before you go to do the photography. Get a flash for your D200 if you can't get sharp pics otherwise in poor light. Get some instruction by someone who knows - an hour of 'hands on' is worth an essay at long distance. Get that flash, it's the only way for starters, spend maybe £120/$160. Later you can learn some fancier techniques.