Since the major U.S. carriers came into full compliance with the CTIA's Consumer Code for Wireless Service on February 11, 2015, they can no longer refuse to unlock your device (assuming you are eligible). This means you can unlock your iPhone and go to a new carrier if you wish.
The big four carriers agreed to come into full compliance with the CTIA's Consumer Code for Wireless Service (section 12) by February 11, 2015, which they all did. This means they can no longer refuse to unlock your device so it can work on another carrier's network.
You cannot unlock your Verizon iPhone simply because Verizon does not lock its 4G LTE or 3G devices. You don't need to do anything to use your Verizon iPhone with another carrier. For more information, see Verizon's Device Unlocking Policy.
Thanks to the CTIA Consumer Code for Wireless Service (Section 12), AT&T, as of February 11, 2015, is required to unlock your iPhone if you are eligible and wish to do so. Once your phone is unlocked, you can switch to a different carrier.
Starting on February 11, 2015 Sprint will "provide any applicable MSL Code" to any customer who requests that their smartphone or tablet be unlocked so it can be used with another carrier. The change in Sprint's unlocking policy was spotted today by Android Police, and comes after the Federal Communications Commission pressured all five major U.S. carriers to give customers the choice to unlock their devices and switch mobile providers as they see fit.
Sprint notes on the company's unlocking FAQ page that "not all devices are capable of being unlocked," and that some devices will not retain all their functionality. For example, all iPhone models and other "devices manufactured with a SIM slot within the past three years cannot be unlocked to accept a different domestic carrier's SIM for use on another domestic carrier's network," according to Sprint. Your device and all its functionality will also need to be supported by the carrier you wish to switch to for it to work on that carrier's network.