What's the cheapest and best way to travel Europe?

I want to eventually travel all over the place. But I’m not sure if I’ll have the money to be staying in expensive hotels in the hearts of major European cities, but I also want to be safe and not end up dead. What is the cheapest, safest, and best way to take a tour of Europe. (or even just a single European country)
Do you have to have a lot of money in general in order to travel to foreign countries? Or is that just a myth?

Is there a way to travel Europe for a year with limited monetary resources?

In about three years I’ll have saved about k U.S. Dollars (life savings) and I would like to travel Europe while also completing my degree. Does anyone know how best to move about freely in Europe while also watching your money? Ideally, I’d like to live in a different city every 2-3 months.