I am a huge traveler (16 countries, all in the past six years), and it can get expensive. However, it's not remotely impossible to do on a cheap budget. My mother knows a woman who travels 4+ times a year on a teacher's salary, and has been to something like 60 countries. You just have to be smart.
http://www.kayak.com is a fantastic way to comparison shop flights. I've used it many times and have always been pleased by the fares I've been able to get. Also, know that sometimes the most expensive flight is from the US to the other continent, and once you are there it can be much cheaper. Case in point: Europe. An absurdly expensive continent in many ways, but I've done both London to Dublin and Madrid to Barcelona for a little over $40 round trip. I kid you not. Baggage can be more expensive though depending on what you are bringing.
Trains and buses in many countries can also be very cheap. I've taken essentially school buses with chickens running around in Central America for dirt cheap...the most pleasant thing in the world? No, but I was able to access some amazingly deserted beach towns because of it, and for very little money.
Accommodation can be cheaper as well. Hostels are often an absolute steal compared to hotels, and I feel you meet more people. Hotels can be brought down by huge amounts using sites like hotel.com. I've gotten some ridiculously nice hotels for peanuts using that site. And if all else fails, there is always
http://www.couchsurfing.org/. This site lets you stay for free in a home stay during your vacation. I haven't tried it yet (but likely will), but my friends have done it and raved. Obviously be safe...look for somebody with a ton of positive reviews.
Last but not least, experiment a little. Paris and London, etc., are fantastic cities, but incredibly expensive and tourist oriented. There are some amazing parts of the world that are infinitely less expensive but just as worth seeing. I've done a ton in Africa, Central America and South America (I'm getting to Asia next!) and found it just as amazing as a European cosmopolitan city (and in many cases, more interesting and vibrant). $10 beach side lobster dinners in Nicaragua? Fifty cent beers in Kenya? I mean, how can you say no? If you still want to do a major country in Europe, pick a different region. For example, the Alsace Lorraine area of France is stunningly beautiful and relaxed, and although it's not exactly budget it is far cheaper than Paris.
Good luck, and have fun traveling!