You want a high ASA which allows for a higher f-stop. The higher the f-stop, the better the depth-of-field. F-stop, by the way is the lens opening. A F-stop of 5.6 allows more light, but has a smaller depth-of-field than f-11. With a greater depth-of-field, a greater range of objects are in focus. That is good for fast moving subjects, like sports, where you don't have the time to get precise focus. However, to do that your camera has less light which needs to be compensated with a lower shutter speed. To enable faster shutter speeds, you increase the ASA (or ISO) which increases the sensitivity of your camera's image sensor. A higher shutter speed, of course, allows less blur from motion (sometimes the blur will add an artistic effect).
You will notice that your 70-300 zoom has somewhere around f-4 to 5.6 as its widest opening. By its nature, your lens has less light coming in than the Nikon AF-S DX NIKKOR 16-80mm which has an aperture between f-2.8 to 4 (Amazon price $1066). I have noticed that these Nikon lenses are quite expensive. I found a Tamron SP 24-70mm f/2.8 at Amazon for $710 which has the same aperture. Make sure these lenses work for your camera.
I knew I got that bass ackwards when I typed it.
Again the perils of posting while working.