Unlock your phone before you travel. (A phone is described as "locked" when a service provider has ensured that it can only be used with a pre-installed SIM card, rather than any SIM card you choose to install.) Paying your provider to give you an unlock code for your GSM phone allows you to install SIM cards from local networks when you are traveling. Some cell phone shops may also be able to do this for you.
Buy a local SIM card in each country you visit. SIM cards, with both prepaid and contract plans, are available from almost any cell phone store. Installing a SIM is easy; just pop off the back of the phone, remove the battery, and slide out the old SIM (a small, rectangular chip with one clipped corner) then install the new one in its place. This will give your phone a local number, and allow you to make calls without paying exorbitant roaming fees. One downside is that your friends and associates in your home country will not be able to reach you at your normal number; you will need to provide a new one to them each time you switch SIM cards.
Avoid using any mobile data functions on your phone. Surfing the internet while roaming is often extremely expensive. Rates vary from provider to provider, but it is not uncommon for the downloading of a single 3-minute streaming video clip to cost almost $40 if downloaded in Europe. If your phone has WiFi capability, use that instead when you can find a hotspot.
Turn off any "fetch new data" features on your phone. These automatically log on to the internet to update your e-mail and contact list, and can run up big data charges if they do so frequently.
Keep your phone turned off when you don't need to use it. If you are bringing your phone abroad with a roaming agreement -- and without changing SIM cards, as mentioned above -- your normal number will still work. Calls from people who don't know that you're abroad could cost you a great deal. Keeping your phone off is a good way to ensure that you don't pick these calls up. You can always turn the phone on periodically to pick up voicemail messages.