This depends on the carrier. I believe that if you switch SIM cards to a new phone without telling your carrier, your phone calls will definitely work without any problem, and obviously wifi does not depend on the SIM card. However, I think that some carriers (like AT&T) require that you register the IMEI # of the phone with them before you can use cellular data, so you *might* need to call them up to let them know the IMEI number on your new phone. I had to do this once when I bought an upgrade phone using my wife's line, but wanted to activate it for myself.
Of course, if you used your phone # when you ordered the phone (either through AT&T or Apple), you shouldn't even need to switch SIM cards. Your phone will have a new SIM card in it, and when you power it on and activate it, the new SIM card will become your SIM card, and the old one will be deactivated.
Two years ago, I ordered an iPhone 5, gave my phone number as a part of the order process, and chose Verizon as my new carrier (I was on AT&T). When my new phone arrived, I just turned it on and registered it, and it immediately started working. I expected to have to jump through some hoops to get the number ported to Verizon, but apparently Apple and Verizon did all of this before my phone even arrived.