There are many different types of travelers. Some only stay in hotels for a week. Others, have $6 in their bank account and will sleep on the beach for months. For the purpose of this, I'll only consider my experience.
When traveling long-term. The main expenses really fall into 3 categories: Transportation, lodging and food/drinks.
Transportation has become nearly free in a lot of places with the growth of bla bla car, public transport in cities and cheap busses. Not to mention flights outside of the US are quite reasonable. Even still, if moving around a lot, this is where the cost really adds up.
To keep cost down on lodging, you can stay at a hostel or couch surf. In most countries, you can rent a room for a month at a time very cheaply as well.
Food is a bit trickier. Prices can very greatly. Look for locations where the locals crowd in. It's likely very good and priced fairly to the location. If you stick around tourist traps, you're going to get gouged.
Prices will range drastically depending on if you're in Norway or Vietnam. I'd feel comfortable traveling on $2,500-1,000/month in those countries respectively. If you're new to travel it may be more but the longer you're gone, the cheaper the world gets!
Tips:
#1 - Find the cheapest flight out of the US, then book a flight from that location to where you want to go. Can save $100s.
#2 - Book multi-city flights with multiple day layovers in between.
#3 - When you first arrive, stay at a hostel and ask the staff a lot of questions. They can help you find long-term lodging, cheap eats and the secret spots to visit.
#4 - Learn: hello, thank you, please in every language and use them.(at a minimum)
#5 - Don't forget - if you need to find a source of income when you return, make sure you have some buffer funds for time to search.