but still with the same network - from which I take it that you have simply moved your SIM from one phone to the other?
That's all you need to do, and the new phone will work immediately it's turned on. The SIM is what identifies you to the network, regardless of what phone it's in, provided the phone is either unlocked, or is dedicated to your network. You can't use a T-Mobile phone with a Vodafone SIM for instance.
If there's something I'm not understanding please provide a little more detail, and we'll try to fix the problem.