Depends on the sport, but for most professional sports, I follow players first, then I root for team.
I realize in this day and age that loyalty is gone. Most teams in pro-sports are run as a business first and look at us fans as mere dollar-signs. They wouldn't donate to a charity without regard for the perception and dollar value that the donation would get from their fanbase.
When I think the team is being honest with itself and what it can afford, and explains a lot to the public, than I will root for them. However, I root for players as they are the product, they are the ones putting themselves out on the line. Players are everything in pro-sports, the entertainment and what not. Granted, they don't do the investing and finances, but they are still the main product that I am buying.
I watch the former Buckeyes and Aztecs with pride and root for them to succeed. I also root for the athletes that represent their communities well, like Drew Brees, Curtis Granderson, CC Sabathia and LaDainian Tomlinson. And then I root for my local team to win. But I am not above putting players ahead of team. Mostly because teams will generally cut a guy who is still productive to the PRODUCT on the field, but costs too much (even though they signed the contract for him in the first place).