I'm not sure you'll get both a camera and lens that does what you want for $600, but I can tell you how I did it...

I have the Canon EF 100-400mm f/4.5-5.6L IS USM, and think it's great for capturing sports and action. Especially on a crop body as it will be the equivalent of a 160-640mm lens.

New, one of these will set you back about £1250, but I got mine second hand on eBay for about £820 (A real bargain!). I don't regret one second of buying it or using it. It's a versatile lens, great for the sort of work you want.

It is however quite a slow lens, so you'll need to marry it with a body with good low-light, high ISO performance. I use it with the EOS 7D, and the two go really well together. For quick moving sports action you also need a camera with good auto-focus, and the 7D again excels at this with 19-point AF, and great AF tracking. It also has an 8fps continuous shooting rate which is ideal for sports as you'll want to rattle off a fair few shots, then pick the best later in post.

Apologies if this is way out of your price range, but believe me, it would be easy to spend a heck of a lot more to get similar shots (the cost of a 400/4 for example). Anyway, hope that gives you some sense of what to aim for...