Flying and Driving
Let all of these miles accumulate and BAM! you have a free ticket.
Second best, although not always available, are buddy passes. If you know of anyone or anyone's parents who work for an airline, there is a good chance that they get tickets for a heavy discount - I'm talking heavy like a Krispy Kreme burger. Just schmooze them up and perhaps they'll slip into your possession.
If, however, you are like most of us and neither of these opportunities are available, then you're going to have to cough up some (or lots of) change. ProBargainHunter has created an extensive list of sites which compare airfares from most major companies, and I can't do much better, so I'll refer you there. However, I will add that, I found Kayak and ITA to have the most accurate and up-to-date search results. I also liked FareCompare the best out of the services that notify you when prices drop.
If you are following the open road, then there is a pretty easy, albeit unconventional way to do it for free (and if you're over 23). You can fill in a form at Auto Driveway and see the list of cars that are available from wherever you are leaving. Your destination may not be exactly where you want to go, but there's no point in a roadtrip if you don't take some random detours, right? They generally pay for most if not all of your gas, and you get to take a certain amount of time to complete your trip, so although it's not extremely leisurely, it's not that grueling either. Beggars can't be choosers, right?
Even more unconventional would be hopping on a train - no, not an Amtrak, but more like hobo style. I've wanted to do this, but I still need to do some more research, and I'll post if I ever get around to trying it out.
Coming up next: lodging
Update: I forgot to mention these two, but they have been making some headlines recently. Yapta (Your Amazing Personal Travel Assistant) allows you to track airline ticket prices. What's more you can download a program, and if you buy a ticket, you can tag the itinerary. If the price of that ticket becomes lower, it will let you know, so you can call the airline and get refunded the difference.
Another new concept comes via Skybus. This carrier flies out of Columbus, OH, and they began offering $10 seats. They are a bare minimum carrier, meaning you basically get a seat, and if you want extra amenities, you have to pay for them, but for the budget traveler, this can be a great deal. Another way they keep their costs low is by going to smaller airports close to the major cities instead of the major airports themselves. I think the verdict is still out on them, but these two are both exciting opportunities to travel cheaply.