Just so happens, I live for this kind of stuff. I've traveled around the states for some time as well. Depending on where you wanna go, and how you wanna get there and your time limit and etc... anyway. Here's some resources:

http://squattheplanet.com/ - Resourceful for free travel

http://www.couchsurfing.org/ - Awesome resource for a free place to stay...but use it actively and often otherwise it won't work at all. Make sure to get referrals and go to meetups if you can!

http://www.hospitalityclub.org/ - An less consumerist alternative to Couchsurfing

http://www.wanderingearl.com/how-i-can-afford-my-life-of-constant-travel/ - Great blog. Read some things, learn some stuff

http://www.reddit.com/r/iwantout - Check out this subreddit as well.

It all depends on how you want to travel the world! A lot of people get jobs teaching English, WWOOFing, you can get jobs cleaning hostels or pubs or what have you a lot. If you're in the U.S...you don't really need a job to vagabond across the country. We throw away enough food and shit for anyone to survive.

You can save about 10k doing a contract through something like Carnival Cruise Lines then travel with that money, do another contract.

If ya got any questions about visas or citizenships, you can ask me that too. Some people will just go to another country until the visitor visa is reset in the country they were in...different countries have different rules. You only have to leave Argentina for like a couple hours, but you have to leave the EU for 90 days.

I mean, really, it all depends on what you wanna do and how you wanna do it. OHYEA! Also, make sure to subscribe to each city's subreddit for meetups and stuff. I met some cool people that way... it's lonely in new cities and it can be daunting at times.

Now I'd quickly like to add that you can volunteer to crew on barge ships in harbors (if you walk around and ask) in exchange for transatlantic or transpacific transportation.

You can get a job at an airline

Many employees get access to cheap or even free flights.

Even if you work for on the ground, in an office or at a subsidiary you can get benefits. My brother worked for a delta subsidiary (pay sucks compared to the mainline operation, and you get a sort of 2nd class flight benefit, but I still get to fly to Tokyo for $47) and calculated the flight miles he and his wife used in one year. They had circled the globe 1.75 times.

Is traveling alone on a budget safe for a woman?

It should be fine if you use basic common sense and not go traveling to places like Syria, or hang out in India or Pakistan in the streets at night alone.

My cousin is a woman and she first went to teach english in a rual area of India and then she ended up in Pakistan, Laos, Cambodia, Jordan, Uzbekistan, Indonesia. She was basically on her own every time and forced to make friends on her own and learn the language and she pulled it off every time.

Now she works for the US state department no doubt thanks to the skills she learned as a world traveler. All this is to say that women can travel the word it just takes a special breed to pull it off.