It took me years -- seriously YEARS -- to figure out how not to pay an arm and a leg to use my iPhone when traveling out of the US. It can cost a lot of money if you don't prepare your phone and your plan properly, so I'm going to share with you what I have learned.
First of all, if you have an iPhone, you cannot live without your iPhone, so the concept of only using wifi while you're traveling is ridiculous. So, don't listen to those fools who say "simply turn off all data roaming unless you have access to wifi!" Those people probably have Blackberries. (Obviously, do your best to find a hotel that has free wifi so that you can surf like mad when you're in your hotel.)
You can use data, but it will cost you a small amount of money (and come on, you have an iPhone already; you can afford to spend $30-$60) and you will need to ration that data. It's really hard, I grant you, to not check Facebook, CNN, Tumblr, Pinterest, Instagram and whatever else you're addicted to every 5 minutes. But I have found -- and I know this is going to sound crazy! -- that not being able to check my phone more than a couple times a day is actually conducive to enjoying my vacation more.
I don't know how it works on Verizon's website, but on AT&T's, under the Wireless dropdown, there is an option to Add or Change Services. Click on that.
Also in that International Features section you can add on international roaming messaging (not international long distance -- that's for sending from the US to other countries), but, if you are lucky, all your friends and family also have iPhones and so you can text them for free from anywhere in the world. If you text other poor souls a lot, you might want to add on that feature for $10/month. Here's more info on the cost of those plans and what it costs to text and send photos via text when abroad.
As for voice plans, I'm not an expert on that since I rarely use my phone for actual phone calls when I'm traveling except maybe to make a dinner reservation or two, so I always just pay for the cost of the call. There is a bunch of info on voice plans, here, though, if you need it.
Step 2: Ensure the international plan covers the whole pay period
This is the tricky part and where I have gotten stumped many a time. When you sign up for 300 MB of international data, to get the whole 300 MB, you will need to apply the plan to your whole pay period. So, if your pay period starts on the first of the month and you add on the plan on the 10th and then you go away from the 15th to the 30th, and you don't backdate the international plan, you will only get 20-ish days of the data plan, ie, 2/3 of the 300 MB that you signed up for. This can work to your advantage at times, but for this, it does not because you want the whole 300 MB. So, you must ensure that the international plan is applied to the whole pay period.
You can ensure that by signing up for the int'l plan in advance of the billing cycle during which you will be traveling. Add on the int'l plan during the pay period prior to your trip and you will be able to select the following pay period when you go to add on the int'l plan. Or, you can call 800-331-0500 and simply ask the customer service agent to backdate the plan for you. I'm not sure what the cut-off date is, but early on in the billing cycle you can backdate the plan yourself. I'm 2 weeks into my cycle right now and I am unable to backdate it on the site. I'll check back periodically and update this if I find out more details on that.
Step 3: Calculate how much data you are using
Just before you turn off your phone when you are on the plane on your way to your fabulous foreign destination, go to Settings -- General -- Usage -- Cellular Usage. Click Reset Statistics. While you are on your trip, check this screen frequently to keep track of how much data you are sending and receiving. It's pretty reliable.
Step 4: Turn off all unnecessary data suckers
Under Settings -- General -- Cellular, turn off Data Roaming. On this same screen you may want to also turn off the other items that are using cellular data, such as iCloud docs, iTunes, etc. (Note: you may need to turn this on again periodically if you need to look something up, but it's best to keep it off when you aren't using it.)
Under Settings -- Mail, Contacts, Calendar, click on Fetch New Data and switch Push to OFF. Then change the Fetch option to Manually so that you have to manually pull in new emails. You can then wait to do that when you are in a location with wifi. And if you need to look something up in an email that you already downloaded, you can open up the email app and look at the email without all your new emails downloading automatically.
Turn off all alerts or at least most of them. I don't want to miss out on the CNN breaking news alerts, for example, so I keep those on, but I turn off notifications that it's my turn in Words With Friends. Go to Settings -- Notifications. Click on the items listed in Notification Center and change to Off.
If you really want to conserve data, you can turn off Location Services located in Settings -- Privacy. Or, instead of turning off the data for all the apps, you can just turn it off for the ones you don't use often and leave it on for the camera, Yelp, and maps, for example. According to the interwebs, the location services itself doesn't use a ton of data, but why not turn it off anyway? That's my philosophy.
That's it! Have a fabulous trip! And relish the fact that you can't disappear into your tiny pocket computer during every moment of downtime. You may even find it to be a wonderfully novel experience reminding you of the good ol' days and you may think you will surely keep this up at home. (You won't, but that's okay. That's part of why we travel.)
Just one more thing. Not to confuse you, but I did mention above that sometimes you don't want to backdate a plan add-on. For example, you may want to add on a mobile hot spot for an hour. You certainly don't want to pay for a whole month's worth of hot spot use. It requires some math or if you're not a fan of math (like me) then you can just turn it off again a couple days later and assume that was plenty of time to include the amount of megabytes you used.