How to effectively detect a rooted device on Android?

There is no general answer to this questions as this completely depends on the type of the rooting procedure used. This paper (WOOT workshop of USENIX security 2015) gives a nice overview of the different kinds of rooting procedures.

So if you want to implement a root checker, your chances are best if you can detect the majority of popular root methods.

SafetyNet Attestation API can do the work for you. It helps you assess the security and compatibility of the Android environments in which your apps run. You can use this API to analyze devices that have installed your app.