If i buy an iphone 5s in usa can i use it in mexico?

For what its worth, I was told by a Company that unlocks phones, that if I unlocked an ATT iPhone 5S that I could buy a micro sim card for it in the Ukraine and that it would work fine.

Don't buy the phone in Mexico, our "unlocking" companies may not be able to unlock it.

You can if you buy an unlocked phone. You MUST buy it directly from Apple, not from any carrier. The only unlocked 5S currently available is the T-Mobile phone, and it is unlocked only if you purchase it directly from Apple and never use the T-Mobile SIM that comes with it.