Can i use a iphone with another phone company?

So for example if i buy and iphone 5s from verizon if i take out the sim card and put a at&t simcard in it could i use the company instead.

If the phone is unlocked, then yes.
And if I remember correctly, Verizon iPhones have their GSM portion unlocked.
You can try this by using a Nano Sim. If you can make and receive calls, then it works.

Now, if you still have a contract with Verizon, or have it financed, you still need to pay them or else…

In theory yes, but of course that doesn't mean the Verizon bill would go away.

Usually though the phone is locked so it would have to be unlocked before trying to do that but the iPhone is sold unlocked (unless that has since changed) from Verizon & also is a GSM/CDMA phone (which normally would make what you are asking impossible)