Could I buy a iPhone in America and use it in Australia?

Could I buy a iPhone in America and use it in Australia? - 1

Added (1). I will buy it from the Apple Store and it will have no sim in it until I get back.

Yes, although some services might not be compatible. In particular, 4G/LTE services tend to vary a lot from country to country.

Probably, if it's unlocked GSM.