Well...sports pics look best when the action is sharp, and stopped, and not out of focus or motion-blurred, so, with those two concepts as a starting point for what you want to come away with, keep a few things in mind:
1) do NOT waste your time shooting at base ISO. START at ISO 400, and work up from there, fast. The D600 has a FF sensor, and fantastic higher-ISO performance, so do not waste it!!!
2) keep the shutter speeds on the higher end of the continuum...think 1/500 as an absolute min-i-mum. The closer the action, and the more parallel the movement is to the lens axis, the faster the speeds must be. The higher the magnification, the faster the speeds must be. So, generally, try for 1/1000 second on most frames. A little faster is better. Depends on the sport, and the exact subject. Depends on the kind of movement, you know?
3) Wide-open, meaning f/2.8 with the 70-200 f/2.8 zoom is "okay", but stopped down just a wee bit, and using higher ISO to compensate, is what I would do under normal circumstances with a modern FF Nikon. A little tiny bit of stopping down, such as from f/2.8 to f/3.2 or f/3.5 will give you better optical quality, and a tiny bit of extra depth of field to make up for slight mis-focusing errors.
4)With a high-quality image to start with, meaning 1)in-FOCUS and 2)with motion STOPPED, you can crop-in later for most sports uses. Yearbook or newsprint or web images do not need to be full-frame, 24 megapixel sized to look good from a modern camera. So, stick with the 70-200/2.8 and use it where it's at its best. The "old" 80-400 VR is pretty slow on AF, but I have used it for soccer a few times...it's kind of sketchy in terms of focusing...an AF-S lens is a better bet.
5) Pay attention to what's going on when you're at the events. On night football, the lights at every HS field I have ever been to sucked. Meaning you'll need ISO 3200, or 6400, around f/2.8 at 1/400, if you are fortunate. MANY field have terible fall-off in the endzones, due to having not really enough lights....it's not like NCAA football...high school fields are often pretty dim. GO for sheer lens speed, and high ISO. Use the 70-200 with NO converter at night.