If you went through upgrade channels, it should be provisioned through the carrier already. Just turn on the new phone and it'll walk you through the steps. You can convert it now in advance if you want but I don't think it's necessary. But definitely upgrade to iOS 16.
If you bought the phone outright then yes, you should probably convert now. Transferring eSIM (from major carriers) is fairly easy. If you have an MVNO, you may have to reach out to have them update your IMEI. You can check YouTube for videos on how to transfer eSIM to new device.