Step1
TRANSPORTATION - AIR TRAVEL
First, you will have to think about how to get there (that is, if you don't already live in europe). Now, flying is never so cheap, especially due to the "bad" economy these days, and rising gas prices for airplanes - which price will reflect on your ticket. So sorry, this part can never be completely free - if you're lucky, it'll be $150-650 usually.
If you're a student, visit www.statravel.com for good deals on plane fares. If you're not a student, there are plenty of sites you could visit for low-budget airfare costs (resources below). Remember to shop around by looking at different websites. It could be a couple hundred dollar difference from site to site. Remember to be flexible with dates, and if you're going to be arriving in one city in europe, and leaving from another at the end of your trip, to NOT book a round trip. One way could be cheaper.
For cheaper (but less flexible) flights, try becoming an air courier. In place of your checked luggage, a company uses it to deliver a package. You must be ready to leave at pretty much any time, and you'll be lucky if you get to travel to Europe (most are for Hong Kong or Singapore or Sydney, etc.)
Another option is to air hitch (www.airhitch.org).
Again, for students (age 18-22), for just $69, you could be travel on a standby flight : www.airtranu.com/airtran_u.aspx
Step2
TRANSPORTATION - INTERCITY (TRAIN, BUS, HITCHHIKE?)
If you're going to be going in and out of countries/cities frequently, then it might be a good idea to get a Eurail pass. They are pretty costly, though, but definitely worth it.
Step3
ACCOMODATIONS
Yes, there are the hostels, that could probably run from $15-25/night at the cheapest, but why do that when you can couchsurf?
www.couchsurf.com is a community where travelers can interact with eachother, and, like the name says, someone offers a couch to sleep on, and you can contact them and ask if you can sleep there. It's totally free, but you've got to be a good guest (or host :)). It might sound unsafe, but if you go with people who are reliable and have a lot of comments from others who have personally stayed with them, it could be a good option. Not only that, but you get to really interact with the locals and experience a part of their culture.
Other free stays include sleeping in airports. check out www.sleepinginairports.net for reviews.
Step4
FOOD
The easiest way to eat cheap in Europe is to visit the local supermarkets, instead of restaurants. If you're from the U.S., get acquainted with the metric system, and the Euro system as well, so you can be sure that you are getting a good deal. Buy staples, like fruits, a loaf of bread, and cheese. Don't buy too much, as when you're backpacking, you're probably not going to carry around a cooler for your cheeses. Buy what you need for the day. A loaf of bread, cheese and a tomato or two could make a couple sandwiches to fill you up the whole day. If you have access to a stove in your host's kitchen or hostel, buy a bag of pasta and sauce mix. (And some cheap wine?)
If you do want to go to restaurants, go during the lunch hour, where it is cheaper than the dinner hours. And try to stay away from tourist-filled cafes or restaurants, where the food is often not as good, or as cheap, as where the locals go.
Step5
SIGHTSEEING
There is an incredible amount of things to do that are completely free, while sightseeing europe. A lot of museums are free on the right days, and then there are the natural scenic ones, festivals, music in the park, etc.
http://budgettravel.about.com/od/destinationsavings/tp/free_europe.htm
http://studenttravel.about.com/od/pickyourplace/tp/free_europe.htm
and that's just the beginning.
do a websearch, "free things to do in --", and you won't ever have to worry about getting bored. In fact, you could technically just pay for the plane ride to and fro, eurail pass, and $10 a day of food, and you could be set.