A common question without a perfect answer for most. That's because all those great sports shots you've seen were mostly shot with pro bodies with fast, long lenses. And from positions that you can't always get to. It really is a combination of those things that drives most great sports photography.
The difference between a D5 and a 300mm f/2.8 and a D7200 and a 70-300mm f4-5.6 can be as much as four stops when all is considered. In low-light or indoor situations where the D5 shooter needs ISO 1600 to get 1/500 action-stopping shutter speeds at f/2.8, the D7200/70-300mm user needs to set their camera to ISO 6400 to compensate for the f/5.6 maximum aperture. ISO 6400 on a D7200 doesn't look as good as ISO 6400 on a D5 to start with.
So, the answer to this question is three-fold:
- Get the best high ISO camera you can afford, and it needs a good AF system. All else equal, FX is going to be at least a stop better than DX.
- Get fast lenses. For indoor work especially, you're better off with a shorter f/1.4 lens and cropping if you have to then you are with an f/5.6 lens. This is an area where you can now make DX about as good as FX: using the Sigma 50-100mm f/1.8 on a D500 is about the same as using the Nikkor 70-200mm f/2.8 on a D5.
- Get long lenses. For the big stadium sports, like football, soccer, and baseball, you're often very far from the action, so even 300mm usually won't cut it.
Thus, about the lowest cost choices these days that get you close to ideal are:
- FX: D750 or D810 body, some f/1.4 to f/2.8 primes (50mm, 85mm, 105mm, 135mm, 180mm), and/or the 70-200mm f/2.8 type of zoom. This still leaves you with the "long” problem. At present there's no great solution here, as most of the more modest-priced lenses are zooms that are f/5.6 or f/6.3 at maximum focal length (80-400mm, 200-500mm, 150-500mm, etc.). Thus, you're probably just as well off adding a TC-20E III to your 70-200mm if you can't afford a 300mm f/2.8, 300mm f/4, 200-400mm f/4, or something even more exotic.
- DX: D5 or D7200 body, f/1.4 primes (50mm, 85mm, 105mm) or the Sigma 50-100mm f/1.8 zoom. You want to stay as fast as possible to make up for using the smaller sensor. Fortunately, once you get into the longer telephoto range, you can use a lens smaller (and cheaper) than in the FX lineup for reach (e.g. 200mm f/2 instead of 300mm f/2.8, 400mm f/2.8 instead of 600mm f/4, etc.).