Can I use an iPhone 5 on all cellular networks?

If I were to buy an iPhone 5 and were to switch say from AT&T to Verizon, would I be able to take that same one over or would I have to get a different version.

In order for your iPhone 5 to work everywhere in the world It needs to be "unlocked". Usually when you buy from specific stores like AT&T or Verizon the phone becomes locked to this provider. My advice would be to purchase an "unlocked" phone from apple. Otherwise if you take a contract with one these providers after your contract is expired you can requested the "unlock code" but contract can last from 6 months to 2 years.