Well you could crank up your ISO. You will get more digital noise but you will be able to use faster shutter speeds.
A faster lens (bigger max aperture) would allow a faster shutter speed.
The rule of thumb is to have a shutter speed that is faster than your focal length. So for example, if you are are shooting at 50mm, you want your shutter speed to be 1/50 or faster (like 1/60).
At least that was the rule with 35mm film...but your camera has a smaller sensor than 35mm film. That gives you the "crop factor"...they say that the crop factor has to be added into the rule of thumb...so for a focal length of 50mm, you would need a shutter speed of 1/75 (1/90 probably).
With a moving subject, you would need an even faster shutter speed but that can depend on lots of different factors. You can try to pan the camera with the movement of the subject. If you do it well enough, you don't need a fast shutter speed at all (if there is enough light of course).
The obvious answer is to use a tripod or something that allows you to fire the camera without touching it. Of course, that won't help with the moving subject...just the camera shake.