One of the advantages of an Android phone is that IF you also use Google apps then everything (contacts, email, calendar, whatever) is on the cloud and synced to any other devices you also have. So, no data needs to be transferred to the new phone and you only have to log into the Google apps with your account info.
You don't mention your carrier or your old phone's brand, but either and/or both (unless the phone was pure android and unlocked) may have put apps on the phone that you were using instead of Google apps. In this case it depends on their backup options as to what transfer methods you have.
