How to take action photos of dogs in the snow?

Ok - i have tried looking this stuff up on google etc but i need someone to dumb it down for me. I have a NIKON D3000 and the lense I have is 55-300mm. I need to know what the Fstop and shutter speed should be on what settings (Shutter - Priority - Manual and Automatic) when i go to Manual the highest it will let me take the shutter speed is 1/200 and the Fstop the lowest I can get it's 5.6 so i need help if there are other ways to get the shutter speed higher and the fstop lower? Lol also the best settings to have th ISO on when your outside taking photos in the snow?

We have no idea what your settings should be because we do not know what light you are shooting in - you can't just 'magic' settings out of thin air! Your camera has a light meter in it - use it!

The f stop is dictated by the lens you have mounted - its likely that f5.6 is the widest that that lens is capable of.

ISO should always be kept low for best image quality unless it absolutely has to be raised to get the shot.

Bear in mind that camera meters meter for a mid-grey tone - snow should be white not mid-grey, so you should dial in some +EV exposure compensation.

For the moment, stick with programmed (Sport) mode (a UV filter might help) and concentrate on following the dogs, remembering to continue the movement as you shoot.

Unfortunately, while the technique sounds simple, it takes a lot of practice to get right - you should end up with a few decent shots if you persevere, though.

Think of ISO setting as the gas pedal in your car, more pressure on the gas pedal the faster the car goes. Higher settings, expressed as higher ISO numbers, make the use of higher or faster shutter speeds possible.
You should use SHUTTER PRIORITY. Just set the shutter as high as you want. But lets say the shutter only goes to about 1/200th of a second, then (to repeat myself) you need to raise your ISO number higher to make the camera more sensitive to the light, and thus gain higher shutter speeds. Forget manual exposure for this. You are just not equipped for that explanation with out confusing the issue for you at this time. That will come with more experience. Your shutter priority will give you the f/stop automatically since you have set the shutter speed where it will stop the action for you.
Photography is a certain amount of experimentation, therefore no hard and fast rules exits for your situation, because light brightness varies all over the place.

I would not use "Automatic" or "Manual" operation. "Automatic may not use a high enough shutter speed to stop the action (movement). Manual is great for a non moving subject but it is impossible to "say well I'll use this f/stop and this shutter speed"-mean while the dogs have grown grey hair and you are a nervous wreck.

Use Shutter Priority or Sports Mode. Set the ISO to 200. On a sunny day it may allow you to use 1/250 sec at f8 which is about where the lens will work best. You could go up to ISO 400 and get acceptable results if you need faster shutter speed. Also make sure the White Balance is correct and the snow looks like snow and without dark blue shadows.
Some cameras will shoot a rapid burst of shots - your camera will do 3fps making it much more likely you'll get a good composition.

I took some pics of our group of Greyhounds running in the snow. Because it was a Fuji camera with a small sensor I wanted ISO 80, and I set it in Sports Mode. (Pic below)

Turning up the ISO will allow you to use faster shutter speed, although it will add more noise. If the light is not changing quickly it could be best to use manual exposure, and do some test shots. You want the snow to be bright enough, but not so bright as to wash out details of contour and texture.
If the dogs are running around very fast it can be tough to freeze the motion. Your best chances of getting decent sharpness is when the dog is at rest between motions. The 55-300mm is a good but not great lens. It lacks the wider aperture of $1000+ pro-level lenses like the 70-200mm f2.8 or the new 70-200mm f4. The slightly fancier 70-300mm AF-S VR will have faster and often more accurate autofocus.

If your camera is limiting you to 1/200 then you must be using flash. If you are using the little pop-up pop-gun flash outdoors then you are wasting your time.

Set ISO to 400 or 800
Aperture priority set to largest aperture available (other than buying a new expensive lens you are stuck with F/5.6)
Snow fools the auto exposure to underexpose so add 1 stop exposure
set to burst mode and fire
If shutter is still too slow then up the ISO