Can I use my iPhone SIM card in another phone (AT&T)?

I have an iPhone 4 on an AT&T family plan. I'd like to purchase a phone outside of AT&T. Can I simply take the SIM card out of my iPhone and put it in the new phone (some no-name-brand Android phone from Hong Kong)?

Alternatively, is there a way to add any old phone to my AT&T account or do I have to purchase one of their phones.

No you must never do that because once you have your phone then won't work

That should be fine, You just need to buy an "unlocked" phone that uses the same size SIM.

You can add any old AT&T phone or unlocked phone to your acct.

You can purchase an unlocked phone or an AT&T phone from somewhere else (Ebay, Amazon etc) and put your sim in it, if it's the right size sim. I don't know if the iphone 4's still use regular sim cards or if they use the micro sim cards. You can also go get the right size sim card from an AT&T store if the old sim card does not fit into the new phone.