I would monitor the traffic on what app store your app is being downloaded to pinpoint frequency of downloads from individual markets, turnover rates from users, and ratings and comments which would contain valuable information from your users. Determining what previously provoked users to your product, what works what doesn't which such a large user base you should be able to derive a bunch of user data to be able to derive a methodology to further study habits of users and pinpoint at least some of the commonalities of good vs bad for your app.
To start you of look into Google Analytics, Fireball, Flurry, they are pretty sold for analytics of small to medium size traffic. All in all whatever method you use please take to account the current privacy issues now in your collection of data to avoid any future complications.