Before you leave for your trip to Greece you should call your mobile carrier to let them know you will be traveling internationally. In many cases, making calls from abroad is blocked and all it takes is a phone call to have this ability activated (usually for free).
Also check the rates you will be charged. Roaming charges add up quickly once you are in a foreign country. It is not unusual for unsuspecting travelers to charge hundreds of Dollars (or Euro) in roaming charges, and that can be a major source of distress upon their return home.
First you should decide if you want to keep the same number when you travel or if you would not mind changing your number to a Greek number when you arrive in Greece.
If you choose to keep your own number, look at your mobile company's offerings and options carefully. Most companies offer discounted international roaming fees in exchange for a small monthly fee added to your bill. Some cellphone companies would require that you use a different devise in Greece.
For example, Verizon, the wildly popular mobile provider of the USA relies on a CDMA network and their cellphones are not compatible with large parts of the outside world, including Greece and the rest of Europe where GSM technology is in use.
If Verizon is your carrier, you should contact them well ahead of time so they can send you another cellphone to use on your trip. Of course, they will still charge you a small fee and all the hefty roaming charges in addition. If AT&T is your mobile carrier, their phones run on the GSM network so you won't need to change cellphones to travel.
While you are on the phone with your mobile provider, ask about mobile data rates, and of course how your calls would be charged (will each call be rounded up to the nearest minute?), and if there are any other fees you will be responsible for.
Once you have this information, you will be in better position to decide if you can afford keeping your existing number, or if you should buy a SIM card (or even another phone) while you are in Greece.
To get a mobile account in Greece you would need to visit one of the ubiquitous Wind, Vodaphone, or Cosmote shops in any large or small town. You need to provide your passport (they are required by law to make a photocopy of it) and an address. Billing your credit card automatically is not an option.
The whole process is rather simple. Within the span of about 15-20 min a service representative will have your account setup (it might not be active until the next day), and you'll be on your way.
If you have your own unlocked cellphone (see below for more details on "unlocked" phones), the service rep will ask you to choose a number from an available pool and will replace your old SIM card with a new one. This will give you a new cellphone number that you can use while you are in Greece.
Besides your new number, you will be asked to choose one of their mobile packages. These are moving targets, but roughly you may opt for a "contract" or a "pre-paid" account. If you are in Greece for less than a month, a pre-paid account is the best option, but for a longer visit you might want to consider a contract that would save you considerable amount of money in the long run.
Once the new SIM is in your phone, keep the old one in a safe place so you can replace it in your device when you leave Greece. Depending on your settings, h aving your old SIM card in a safe place means that all your old data will probably be out of your cellphone - and that includes your contacts. Just because your old contacts are not available after you insert the new SIM does not mean they are erased. They are still stored in your original SIM card, so don't panic.
The service rep might, or might not be able to move your contacts to your new SIM. If you are phone savvy, poke around your settings and move all your contacts to the phone's external memory card from the SIM (there might be a menu item for "back up your contacts to the SIM or the storage card). You should do this before you get to the store so you don't delay there.
If you have a smart phone and need data access on your device, you may add it to your Greek mobile account. Read more about cellphone data here.
The above paragraphs describe how you can get a local Greek mobile number for your "unlocked" cellphone device. This is a critical detail.
Mobile companies in the USA lock the devices so they cannot be used with another carrier's SIM card. If you bought a subsidized cellphone from AT&T or T-Mobile (Verizon and Spring customers don't have the option of using their device outside the US without roaming), then your cellphone is almost certainly locked. This means you will not be able to simply put in a new SIM as described above.
You need to ask your mobile provider to unlock your phone. In most cases, they will decline, especially if you are in the middle of a long contract, and they will ask you to use their roaming plans.
In this case your options will be: roaming or getting a second cellphone device while in Greece.
This is not as radical as it seems. Most mobile providers in Greece offer pre-paid packages that include an inexpensive cellphone. In this case you'd be carrying two devices with you, but this might be cheaper than getting charged roaming fees.
This is also a cleaner solution since you won't have to change anything on your original phone and you can still use it in an emergency (or to look up your contacts and other data, or to use it in wifi hotspots). The drawbacks include the extra expense for a phone you will have no use for after your Greek holiday, and carrying around a second cellphone.