@cvigano First recommendation: Don't set up DNS server, but use some service with nice API. Those are really cheap and lot easies to manage. I myself use AWS route 53.
Second, just copy all your DNS zone data to new service and then make sure it responds to the queries the same way your current DNS does. Then wait for 24h before changing name server info in your domain registrar. Then wait another 24h before changing anything else.