Purchasing unlocked phone

So I have been a customer with ATT forever (seriously) and I currently own the Lumia 920 Windows Phone, I can't stand this phone. We all know HTML5 is best and flash on Andriod is no longer supported etc. - but no one is changing their websites and no one is creating apps for this windows phone  - so you are extremly limited as to what you can do with the phone - pretty much only text, speak and visit websites with no bells or whistles and that's it, at least with an Adriod there are more apps than you will ever need and with Puffin and Dolphin browser able to view flash - anyhow /rant off (sorry it is frustrating).

Anyhow - I am going to purchase an unlocked Huawei phone which says it will work with the 4G/LTE ATT network.

And I just need to double check...am I only removing my sim card from the Lumia 920 and putting that sim card into the new unlocked phone and I am good to go and it works like normal - or do I need to call ATT and let them know anything??  I of course already have the same data/text/talk plan that I have had for many years.

Oh - also - I still have my old Galaxy S phone, while I wait for the new unlocked for to be delivered, am I able to just put my sim back into my old phone and use that for the time being, or would that require a call to ATT also?

thanks!