You go to the new carrier and they do it for you. You do not need to do do anything with VZW.
Yes the new carrier should have done it when you bought the phone so you would just get your old number working on the new service with in 24hrs or that's how it worked when I ported my Verizon number to another carrier. I was told to carry both phones on me as the Verizon one would work for a bit then once the port happened the new phone would be working. Mary