Adjusting iso with shutter speed?

I bought a nikon d5200 with 18-55mm kit lens recently, and i'm having a problem adjusting the iso with the shutter speed fore example i want to shoot my cat running inside the house if i want the light the image comes too blurry and shaky, if i want the photo still, its quality is very bad, and when i use the flash its too slow and won't allow me to go higher than 200 in the shutter speed also the cat close its eyes and stop running… All i want is a still photos with a good light and have some quality.

You need light to create an image. To get enough light into the camera in low light conditions you need to slow down the shutter and/or open the aperture and/or increase the ISO. Slow shutter gives blurred image, wide aperture gives shallow depth of field, high ISO gives speckly image. 4th option is to provide more light - use a flash, shoot outdoors, use photo lights.

To shoot without flash - set your camera to aperture priority mode, open the lens to widest aperture, set your ISO to 1600. The camera will select the shutter speed.

With flash the shutter speed is largely irrelevant. All you need is to keep it open long enough for the whole frame to receive the 1/10,000th second burst of flash.

Then you need to take the photo in a location with more light. Cameras are not magic… If there's not enough light, you will not get a good photo.

First of all, with the flash a shutter speed of 1/200 will be plenty fast enough to "freeze" the movement of the cat, you do not need a faster speed. Without flash, you are banging your head against the wall of fast shutter speed / high ISO. Yes, for best quality, a low ISO is better, but if you want / need the faster shutter speed, then you have no choice except to use a higher ISO… Or… Get a lot more ambient light into the room. Also, a faster lens would allow you to set a faster shutter speed at a wider aperture with less ISO. A lens such as a 50mm 1.8 would help you a lot, but it is not a miracle worker. Photography requires light, and it is all a balancing act between shutter speed, aperture, and ISO.

Your first mistake was in thinking the cat would cooperate with you in any way.

ISO and Shutter Speed are two parts of the Exposure Triangle. The third part is Aperture. By learning about the Exposure Triangle you'll understand how the three parts work together to produce an exposure.
http://digital-photography-school.com/learning-exposure-in-digital-photography

Another problem you have is trying to use the slow 18-55mm f3.5-5.6 zoom indoors without the flash. In this situation your only option is to increase the ISO until you get a shutter speed of at least 1/250 sec. At f5.6. Of course, increasing the ISO results in more digital noise but without using the flash its something you have to live with. A program like Noise Ninja can help with the digital noise.

When using the flash your shutter speed is limited to 1/200 sec. Or lower. 1/200 sec. Is the flash sync speed and trying to use a faster shutter speed will result in only part of your sensor being exposed. However, using the flash will "freeze" the movement of your cat. Try using a piece of white tissue paper or thin white cloth to diffuse and soften the light from your camera's flash.

NOTE: Do not get too close to your cat's face when using the flash. Cats see things about 4 times brighter than we do and a direct flash could damage your cat's eyes - even diffused with tissue paper or cloth.

So learn about the Exposure Triangle and either invest in a Nikon AF-S 50mm f1.8 prime lens for indoor use or plan on using a high ISO or use the flash.

To be honest if you just set everything to Auto or Action (if your camera has that feature), you will probably get reasonable results. The camera can do all the thinking for you.