My app has been in the App Store for 2 years and is top of its niche. We launched on Android 8 months ago and this is also going well. I want to take the app to the next level by learning from how my users (over 10k users per day) use the app. What do they press as soon as they open the app, why do they uninstall (or what were they doing right before uninstalling), how does a brand new user use the app (i.e. does he go straight to our tutorial or just dive in). Any suggestions on SDKs, websites, etc . . that would help me to track down and study this information would be appreciated.
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.