The 70-200 with a 1.4 and a 2.0 convertor will work just fine, a friend and I run around here where I live and we shoot the high school games, he uses a 80-200 and I use a 70-200 and I have never felt like I was lacking, I at times have also used my 300 f/2.8, but it so much heavier, that it rarely sees use anymore, and we have been doing it for almost ten years now, the 70-200 is handhold friendly and lets you shoot quickly, you didn't mention what type of camera system you use, I shoot with a Minolta 9xi with a 4.5 FPS motor drive, which allows banging off frames pretty quickly when needed..

I would not even consider a 400 f/2.8, that is a monster of a lens, that is cumbersome to shoot even under the best conditions..

Have fun, head to the nearest local park and take images of people at play and you will find your sweet spot that will work for what your wanting to do.