I bumped into footman at Bushy Park last week with my dogs and we discussed this post and I said then it is difficult to get good shots of dogs running especially if they are quick. I dont get on here much but here is my take on what to do.
You will get lots to throw away because dogs will change direction, dont go where you expect them to go, they will run behind bushes, into shade or away from you and walk back. Unless they are on a track they are unpredictable and even then they are hard to get as they are quick.
If its photos of your dogs you want before you worry about camera settings think location. Try and find a good spot to take the photos. An enclosed field or open space without loads of people or other dogs to start with. If your dog chases balls get a ball thrower, get someone else to throw the ball and position yourself to the side of where the ball will go for panning shots or in front for running towards shots.
Put your camera in (C) Continuous tracking mode or AI servo (Canon) and select single focus point and auto focus on, maybe multi frame 3fps on your camera. You will need to control your shutter speed so select S mode or Tv (Canon). I shoot in Av but to make it easier stick to S ot Tv.
Panning shots. Switch off image stabilizer on your lens, set the shutter speed to 1/160 or 1/200 to start but change to suit the speed of the dog after taking a few shots. You want motion blur but the dog in focus. If it is bright set ISO to 100 or increase if gloomy to keep the speed selected (or auto if you want). Think where the dog might pass and set the focus length to suit but keep it wide as you can crop later. Dont be too close to the action. If you have a rear focus button use it, practice moving in a smooth action. Focus on the eye/head of the dog and track it. If you have an all black dog nows the time to trade it in as you camera will struggle. If the dog has passed you stop shooting. You want to get it just in front or alongside. This is not easy with quick dogs I can tell you that.

1/160 f4.5 ISO800 as it was gloomy. Not the best example but the last one I took focus is on the rough coated dog as she comes up to give Ollie a nip on the neck.
Running Towards you. Again IS off. Here you will need more speed to freeze the action so start at with 1/800 and go from there, ISO high enough to get this speed. Again zoom in or out to leave room for a crop or straighten. Get down low kneel of even lie down you then get a better perspective and a greater feeling of speed. Focus on the eye again you could expand the focus points if you want here. If the dog is too close to you it will blur out if your aperture is open so try to keep some distance from the dog and crop. Some slower focusing lens/cameras might struggle here if the dog is quick (my old Tamron 18-270 did).

1/800 f6.3 ISO200. We have radar gunned Daisy at 38mph on a short track and here she is flat out and 1/800 freezes the action enough IMO.
Practice and practice some more so if you want to borrow a couple of lurchers every day for an hour or two just ask.