You use your normal phone. Get your connection provider to unlock your phone. Some providers are reluctant to do this, especially in the first year of a contract, but if you run into problems just persist until you get your handset unlocked. Find a local supplier of 3G SIMs. Purchase one, and if necessary purchase internet access on the SIM card. Insert the SIM card, and initialise it according to the sellers instructions.
Ease of use
Is very variable, in part depending on the state of the market in the country you are in, in part your grasp of the language in use in the country you are in; the availability of speakers of a language you recognise; etc. At best it goes very well; at worst it can prove impossible to use this route.
The specifics vary from country to country. There are different ways different providers provide a service. The easiest I have found is Beeline in Russia - you just plug in the SIM and everything works. The most difficult I have found is a combination of a China Mobile SIM and a HTC Wildfire phone. These seem determined not to talk to each other and all my efforts to make them talk have come to naught. The process seems to involve: setting up internet access on the SIM card, by phoning a help desk number and getting an access code that you send as an SMS. Then it seems to involve enabling data on the phone, before each use. This is where I have failed to get communication and I have no idea why or what to do about it.
Instructions and SMS messages will come in the language appropriate to the place you are in. This is not too much of a problem if the country uses a roman script, because telecomm terms are likely to be recognisable no matter what language is in use. However, if different scripts are being used - Chinese, Arabic, Cyrillic - it can be impossible to recogise instructions to send an SMS or call a specific number.
SIMs, in those countries that permit open purchase of them, are readily available at airports, telecomms providers shops, and other outlets. Some countries are very restrictive in who can purcase SIMS - requiring residency requirments for instance
Also other countries do not use SIMs (Japan), though I have no personal experience of these countries, and don't know what they use as an alternative.
To purchase the card, and set it up so that it can be used can take a lot of time and effort, depending on country, location, carrier and phone.
You will be subject to local law and restrictions. So in China, behind the Great Firewall of China, you will not be able to access facebook, ebay, blogger, and many other sites. Sometimes the entire internet will vanish.
This is by far the cheapest and recommended way to go. There are still a data rate to be paid but these rates are likely to be very low (and may indeed in some places be zero). There is still one gotcha to be aware of. What constitutes local. In large countries (Russia, and China) each major city is likely to be considered local. So, for instance, a card bought in Beijing will incur data roaming charges if used in Xining. These data roaming charges can be very reasonable, within one country, or they can be at the level you would incur if you were using your home SIM.
This is a special note for Apple users. It is possible to gain all the advantages of this approach, but availability of SIMs that are in the Appple form factor is very low indeed. I have seen an Apple user get all the advantages of this approach by conducting very exact surgery to cut down a normal SIM to Apple's form factors. If you choose to go this route, be prepared for failures. The person I saw doing this bought three SIMS, cut down the first, which did not work; and was only successful with his surgery on the second SIM.