It can be used on any carrier, assuming the carrier uses the same signal.
Verizon/sprint uses a completley different kind of signal called CDMA vs ATT/T-Mobile/Pretty much rest of the world called GSM. Not to mention that there are over a dozen different LTE bands that different carriers use and most phones only have a portion of those bands so in short there is no such thing as a universal phone that is compatible with ALL cariers.
Doubt you are going to Verizon because it is cheaper then AT&T so the cricket suggestion is not really viable if this the case.
So bassically you are going to need to sell your old phone and get a verizon iphone.to switch to verizon.