AT&T iPhone4 SIM to iPhone5

I have an iPhone 4 with AT&T for many years now. I am thinking of buying an unlocked iPhone 5s , mostly because the home button on the iPhone 4 is not responding well.

Can I just get the unlocked phone from Apple and use the SIM from my iPhone 4? That is, will phone service just continue when the iPhone 5 gets the older SIM or would I need to contact AT&T and start a new plan/change the existing plan?

Will an unlocked iPhone 5s be useable with Verizon or AT&T (with the appropriate SIM I guess)?

Thanks,
David