Realistically, almost every career offers some travel possibilities. Sales personnel attend training sessions. Professionals attend conferences. You can be a computer specialist in Belgium as easily as you can be a computer specialist in the U.S.
My emphasis here is on jobs where travel isn't just a fluke, but an actual job requirement. Land one of these jobs and you will see the world ... though possibly not at your convenience or on your terms!
(Want to travel when/where it's convenient for you? Marry well or win the lottery!)
- Airline Pilot/Steward. The classic choice for people who want to see the world, for good reason: these days, there's almost no exotic destination not serviced by air. Before you sign up, however, consider carefully whether you have the "right stuff" to endure endless hours of airport muzac, disarm shoebombs, and clear away sick bags with a smile on your face.
- Seaman. It's an oldy but a goody - become a sailor and see the world! Or at least those parts of the world that have ports. You can join the Navy, Coast Guard, Merchant Marines, or become a seaman for private merchant vessels. (Just remember your vitamin C tabs so you don't catch scurvy.)
- Military. At the time of this writing, various U.S. military services maintain bases in such exotic locals as Japan, Germany and the Philippines. The downside is that the parts of the world you're most likely to see are Iran, Afghanistan, and war zones being policed by U.N. forces.
- Cruise Ships. The cruise ship industry is always hiring: the bad news is that, in general, they prefer to hire internationally since foreign workers are so much less picky about matters such as work hours, salary and accomodations. However, there are opportunities for U.S. citizens, particularly in the areas of cruise staff, day care staff, and entertainment.
- Diplomatic corps/State Department. This is an obvious choice, but be prepared to take what you're given - the State Department has a pretty ingrained policy of alternating so-called "desirable" postings (places with beaches, orchestras, history and/or really good local breweries) with undesirable postings (war zones, 3rd world countries, areas of tectonic activity).
- Peace Corps/Volunteerism. Then again, if you're an adventurer, a risk-taker, a romantic ... if you like the idea of spreading peace and doing good ... if you see "undesirable postings" as a challenge rather than a punishment (and if you aren't to particular about the quality of your health care) ... then the Peace Corps may be for you. Failing Peace Corps, there are plenty of other international volunteer organizations willing to whisk you oversees in exchange for free labor.
- Translator. If you are fluent in one of the "hot" languages (Spanish, Chinese, Arabic, etc.), companies may be willing to pay you to travel to these countries to help their sales representatives conduct business with the locals.
- International Business. Or, you can be the guy that the translator is translating. International companies engaged in law, banking and sales need employees willing to travel to oversee/train/manage their foreign operations.
- Nanny/Au Pair. This is an obvious choice for young women who don't mind working with children. The problem is that families will expect you to stay for more than a few months, so you may not get to see many countries before the demands of nannying wear you out and leave you begging for home.
- Student Laboror. In countries with manual labor shortages (notably the U.S. but also some small European countries with tourist seasons), companies often spring up that specialize in importing students from other countries to fill temporary vacancies. While a viable option, expect to work long hours and live in intimate quarters.
- International Courier. Some companies will pay good money to ensure that sensitive materials travel safely from country to country. You'll need to be bonded, insured, cleared, fingerprinted, and you may have to turn over a child or two as hostage, but in return, people will pay you to cart their precious documents, diamonds, etc. from destination to glamorous destination.
- Travel Agent. Though the practice isn't as common as it used to be, resorts/hotels/cruise ships/etc. used to offer low-price or comped trips to travel agents in hopes the travel agents would then recommend their facilities/services to their clients. I suspect, however, that if you can convince these hospitality providers that you influence the decisions of a big enough client base, perks are still available.
- Travel Writer. I have a feeling this probably isn't a lucrative career - if there was big money to be made from travelling and then talking about it, hard to imagine why anyone wouldn't do it. Rather than set out on your own, perhaps the thing to do is to offer your services to one of the going travel writing concerns: Fodors, Rick Steves, etc.
- Hospitality. Hook up with a mega-hospitality provider for a chance to work at hotels, spas, restaurants or resorts overseas. While outfits like Marriott, Hilton, Sheraton and Intercontinental are more likely to land you in exotic resort areas, less glamorous companies such as McDonalds, Starbucks, and Walmart are aggressively expanding across the globe and may offer more immediate opportunities.
- Tour Guide. Offer your services to a travel company that specializes in offering guided tours to foreign parts. This works best if you have some special expertise to share - fluency in a language, expertise in a particular era of history, or specialized athletic skills (white water navigation, climbing, parasailing) - but some companies may be willing to give you a script to memorize and set you on your way, if you show yourself capable and game.
- Professional Sports. Certain sports - such as golfing, surfing, and scuba diving - require you to travel to where the best facilities/conditions are. But even more mainstream sports like baseball, basketball, and soccer enjoy worldwide appeal, with franchises in almost every continent. If you aren't born with natural athletic ability there may still be opportunities in sports medicine, training, and coaching.
- Academics/Research. Positioning yourself as an expert in pretty much any academic area will provide opportunities for travel - speaking at conferences, guest-lecturing at foreign universities, conducting research in international libraries, etc.
- Academics/Teaching. There are always opportunities for folks able to teach English to citizens of foreign nations eager to learn the "international language of business." Potential employers include governments, embassies, foreign companies, and foreign universities/schools.
- Science. Some scientific disciplines - such as geology, biology, and mechanical engineering - practically require travel as a prerequisite. After all, if you want to study rocks, plants, or dams, you have to go to where they are, right?
- Arts/Entertainment. Cultivate an artistic talent and, if you're good enough, you should be able to market yourself in any country that enjoys drama, orchestra, and/or dance.
- Journalism/Foreign Correspondent. A gig that's as hard to get as it is glamorous. Just know in advance that they're not going to send you somewhere unless there's actual news to report, so you're far likelier to end up in Iran than Belize.