I think teams would be split into varsity and junior varsity where eight people make up the varsity and everyone else filters into junior varsity. Varsity matches would be full six on six battles where points would be determined by how many pokemon the winning trainer had left in his party. The team that won would be the team that the trainers racked up the most points overall. Junior varsity would go the exact same way, except that the battles would be three on three.
I turn to my tennis background for how districts, regionals, and state tournaments would go. The trainers would get separated into seeds and their school (ex: school A #1, school B #5, school C #1) and then the tournaments would start with six on six battles until we have an overall winner.
That would be the competitive side of battling, while the recreational side would be like intramural leagues (I think that's the right word) in colleges.
Tennis once again holds the answer as we look into the professional life of this sport. There would be giant tournaments where trainers from all around would come and compete for the first place prize. Much like how the rankings in tennis goes, how well you did in tournaments, various matches, etc. would determine your world-wide rank.
Related to the games, I think there would be a network of gyms in each state, and then a bigger network for the entire country to determine the champion of that country. Once in a while (like at the Olympics perhaps?) all the countries and their champions would battle for the title as Pokemon Master.
Yeah, I've thought about this before...