do you want this only for dog action shots? good daytime lighting? sometimes at night under lights? How close will/can you be?
If I could be fairly close and have decent control over how far I'd be, I'd go with a 50 or 85 prime. very fast focus, sharp pictures, and way under your 600 dollar budget. If you add a bit a 135 prime is even better, but you're looking at 700+ for a used one probably.
I don't know much about zooms other than a 24-70 is nice, but for different purposes I think and much more. Beyond that I'd be looking at a 70-200 2.8 is my next purchase, but those are also considerably more.
I'm no expert by any means, but i've found for the money, you can't beat the optical quality of those 50, 85, 135 primes and focus for action shots. The rest might give you "more" in variability, but they lack in focus speed and sharpness or low light function, etc.
But others have different priorities than me or the person next to them. So you really have to decide exactly what you want the lens for.