Different options may be cheaper at different times and in different places, so do your research!
Flying can be quite cheap in Europe with many discount airlines – Easy Jet, Ryan Air, Wizz Air, Air Berlin, Monarch, Vueling, Air Europa, etc. Sometimes you might also find good deals on Alitalia, British Airways and other established airlines. Keep in mind that discount airlines are generally no frills, luggage costs extra, and often operate out of more remote airports, which can mean long (and possibly expensive) travel times into the city centre. It might be worth paying more for a traditional airline at a more convenient airport. It goes without saying that including stopovers in your flight often results in cheaper costs – but again, need to weigh the true value of this. You can also maximize your stopovers to extend your trip and see a new place. When flying to Africa or the Middle East, spend a day or two in Amsterdam. When flying to Australia, stop in Fiji or the South Pacific Islands – it can be much more expensive to get to these places from North America directly than to work it into your existing trip.
Another tip is to use the website setting as though you are a resident of the country in which you are travelling to (e.g. if looking at internal Air France flights, search http://www.airfrance.fr/ rather than your home country website). Use local travel agencies as well. North America, Australia and Europe offer inexpensive All-Inclusive packages to nearby hot spots like the Caribbean / Mexico (from North America), South Pacific / Asia (from Australia) and Africa (from Europe) that can be purchased locally. In Europe I have used TUI to book a holiday package to the Canary Islands as an add-on to my European vacation. Thomson is another good one if you are using the UK as your base.
Trains can also be an inexpensive and nice way to travel. There are several options in Europe, but my best deal is the Bahn overnight from Amsterdam to Warsaw for 29 Euros — option to get off in Berlin as well. You could never fly for that price – plus this way you get to see a bit of European countryside and save a nights accommodation (the seats recline).
Driving is a nice option when it’s about the journey and not the destination. Renting a car can be a pain with all the hidden charges, insurance mark-ups, etc etc. This year we discovered Relay Rides in the U.S., a website where you can rent cars from regular people (sort of like AirBnB but for car rental). I am in love with this idea!!! We rented a fully loaded 2013 Jetta for a week, took it from LA up to Napa/San Francisco and back, for about $170 USD — an incredible deal! They do daily rentals as well. You don’t need to be American to use this website, just leave enough time before your trip to obtain, submit and verify your documentation online (e.g. driving record).
You also have the option of letting someone else do the driving for you. In Africa, we took an Intrepid Tour from Johannesburg to Durban, which covered 3 countries and enabled us to sit back enjoying the scenery without any worries about how to get to our next destination/figuring out timetables/etc. The disadvantage of this is that you do have to follow a set schedule, but if you want to cover a lot of distance with relatively little hassle in an area where you are not very familiar, this might be the way to go.