There are several utilities to help protect privacy on your phone, provided you have root access. Along with their blocking functions, some have logs or give you instant info about what an app is doing.
XPrivacy runs on the XPosed framework. It is designed to let you allow, deny, or spoof tons of different android framework calls. For example, you can block a specific app from getting access to your contacts or from acquiring your GPS location.
With recent versions of XPrivacy, the app pops up a dialog box with a 15 second timer (just like a superuser request prompt) for any permission you haven't granted yet. The first time you open an app, it will prompt you for multiple things, but then the settings are remembered for that app later. By default, it only starts asking you about apps you install after it is installed (so that you aren't overwhelmed with dialog boxes the first time you reboot and all your apps start asking for things).
XPrivacy is a great tool for privacy (hence the name). Unlike Permissions Manager, XPrivacy does not cause an app to crash when a requested permission is spoofed/denied. However, XPrivacy is designed specifically for privacy, so it doesn't cover all possible permissions an app can request.
AFWall+ is a firewall, designed to allow or deny network access to individual apps based on what type of network you're connected to. For example, you can block a specific app from ever using cell data, while still giving it wifi access.
Permissions Manager lets you deny specific Android permissions that are requested by apps. While similar to XPrivacy, it differs in that Permissions Manager will outright block an app from getting a permission; If the app isn't properly designed to handle a lack of this permission, the app may crash. Also, it only blocks permissions that are requestable, whereas XPrivacy gives you deeper and more fine-grained controls.
There are plenty of other apps designed to give the user control over the apps installed. The ones I listed here are ones I use extensively or have used extensively in the past.