I haven't done this yet but I did contact Microsoft Support about transferring over to an iPhone (it's mostly the same for Android).
- For stored data on phone (Music, Videos, Documents, etc.): Upload Files to OneDrive or Transfer them onto your PC. Later transfer it to your new Android or iPhone.
- Install the Microsoft Apps from the Playstore or AppStore to your new phone (Mainly Outlook, OneDrive, Cortana, ToDo, Edge, and Office if you use Office).
- Add your Microsoft Account Email to your new phone as well as sign into any Microsoft Apps with it. If using android Microsoft recommends the Your Phone App.
- Contacts: Ok this one I don't think can be done on Android yet without reentering everyone because of all the different OEM Default phone/contact apps. The People App isn't available for IOS or Android, but for IOS you can Sync your Contacts through iTunes on your PC.
Helpful Links I was given:
However I have Office 365 which is how I got support for this when I called, so I'm not sure if this is the same without Office 365 or impossible without it. The third support page sent link sent to me was for Outlook Exchange, and my Lumia uses Exchange ActiveSync.