Auto only covers the shutter time and f stop. If you want less blur with motion, you can either add more light (reduce the f stop) or increase the sensor's sensitivity to light (increase the ISO). For fast action like sports, try ISO 800.
The problem with high ISO is grainy photos, so don't go to high to fast so you can find what looks good to your eye.
If you get grainy photos at low ISO (under 300) or you are still getting blur with a high ISO (over 1600) then you need to reduce the f stop. This can be done with Aperature Priority mode. The lenses you use will determine how low the f stop can go.