What type of lens would you use for a moving shot?

I have a Nikon D3000 and I really want to be able to take pictures of my horses and dogs running and not have them blurred (like this ) So I was wondering what type of lens would I use? And what type of lens lets you take multiply shots at once?

The 18-55 you already have is more than capable of producing images like that…

It is not the lens it is the shutter speed. The horses are moving while the shutter is open and the camera is recording the image.

If the shutter is open too long then the movement of the horse is obvious- resulting in the blur.

There's no right shutter speed, it depends on how far away from the horse you are, how fast the horse is galloping. I suggest starting at about 1/500 and experiment, if there's still motion blur then use a faster shutter speed.

http://aviewfinderdarkly.com.au/2011/03/08/using-panning-in-motorsport-photography/

You can use any lens. What will matter is the shutter speed. (although, that picture looks pretty static; sometimes a tiny bit of blur in the legs looks better than a completely blur-free picture)