If you're a Verizon or Sprint customer, you're in luck. Your iPhone has a card slot, but in all likelihood that slot is already unlocked. (If you're not sure, call your carrier and ask.) In the U.S., your phone doesn't use that slot—it's only there for international roaming.
Buy a prepaid SIM card
Most of us have gotten used to having freely available Internet access in our pockets all the time. It's disconcerting when it's not there. To keep using the Internet in the way you're accustomed, seek out a local cellular provider once you arrive in your destination country, and buy a pre-paid SIM card. (And make sure they've got the right size of card for your iPhone; newer iPhones use nano-SIMs, while older models use micro-SIMs.)
When I recently went to Ireland, I visited a Three store and bought a prepaid card for €20. I was in and out of the store in five minutes, about half an hour later my iPhone was on the local 3G network, and I had unlimited 3G data for my entire stay. I was shocked at how easy the entire process was.
Offers differ from country to country, so it's worth doing a little research before you travel. I was able to identify the right carrier, the right deal, and even the location of the store I needed to go to before I had even left the States.
Keep in mind, for this to work your iPhone needs to have an unlocked SIM slot. I used a Verizon iPhone 5 and it worked just fine, and it worked in an unlocked iPhone 5s as well. Also, once you switch SIM cards, you won't be able to receive calls at your old phone number until you switch back: Your phone will have a new, local phone number. (Internet-based stuff like iMessage and email will all still work as normal, it's just your phone number that will have changed.)
I only ran into one problem with this approach: I tried to log into my Facebook account, and it detected that I was logging in from an unusual location (Ireland). It wanted to do a two-factor authentication to verify who I was, by sending a text to my U.S. cellphone number. For that to work, I'd need to switch back to my old U.S. SIM card.
If your iPhone's not unlocked
If you're on AT&T and your phone's not at least two years old, you can't buy a prepaid SIM card. Do you have a cellular-enabled iPad or (even better) iPad mini? Those devices are already SIM unlocked, so they should be able to accept a local SIM card just as easily as your phone. Toting an iPad mini around foreign streets might seem a little silly, but it could be a nice workaround if there's just no other way.
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