Will apps and also wifi work on on your new iPhone if you don't yet have a phone service carrier?

WiFi should work. Apps that don't require a mobile number should work. If I recall correctly, some third-party messaging apps such as WhatsApp do require a phone number.

What you won't be able to do is activate iMessage or Facetime. Activation of those features require that the phone be able to send an SMS message which is only possible once the phone is activated on a carrier.

After the iPhone is activated the iPhone will work all its functions except for making Voice Calls, SMS, and MMS.