Best thing to do is just read the steps on the following link: http://www.kingdomsatwar.com/transfer/ Which is the official KaW website, so that should make you able to get it done. Good luck!
1. Have account linked via Facebook or ATA ID 2. Download Kaw on new phone, login with the info. Your account is now on your new phone 3. Optional: delink from other phone unless you are getting rid of it then I would delink for incase