I only follow a few sports teams, but for the teams I do follow, Sportacular (that's the iTunes link for the free version, and here's the paid version, sans ads) is definitely my app of choice. For nearly every professional sport you can think of, the app will track scores, plays, schedules, standings, and anything else you need to know about everything from NFL to college football to the English Premier League and the PGA.
Right after iPhone OS 3.0 came out, the developer added in push notifications, and now I've got push updates coming whenever the Cubs have a game starting and ending (or at least, when they did -- there's always next year) and whenever a Bears quarter ends. Sportacular even has a few social features happening -- you can make picks and discuss each game with other fans using your Facebook login, as well as read news and updates about each team in the games you check out.
The app was updated to version 1.6 last night, and the new features include better game pages for NFL and NCAA football, and option to manage and track your fantasy football team in conjunction with their fantasy football app on Facebook. The one bummer I have is that they don't have actual media of the games -- for that, you'll probably have to go for an official app. But especially as a free download (the ads are definitely non-intrusive), Sportacular is really impressive with everything it does to help you track your favorite sports and teams.