Christine,
I chucked it all and left the States but with my boyfriend and with a good safety net and some planning. Definately plan ahead and whatever money you think you need, you'll use about double. Think about if/where you are going to work, where you will live, the visa situation in each country you want to go to. It's fantastic to dream of travelling the world and all the fantastic people you'll meet, but then reality sets in and you'll be unhappy if you didn't plan for it.
I can't say much about the rest of the world, because I went there with the government, but as for most recently going it 'alone' so to speak, Thailand is a nice place to start. It's relatively easy to find work on Ko Tao, albeit illegal work, but everyone does it. Helps if you are a DM or Instructor but you can also work at a shop and learn at the same time. Accommodation can be cheap depending on your requirements. You'll meet a lot of other travellers. Really you can even start on Khao San Road in Bangkok, lots of young single travellers. There's also a city in the north of Thailand like that, Pai. A TEFL certificate will get you a long way in Asia, you can teach English and get visas in most countries.
Anyway, whatever you choose to do, please do your research and make sure you have plenty of money and a cut-off point when you will go home. I have seen people here that keep thinking they will make money and they don't and spend the cash they should have used on a plane ticket home then can't even afford the departure costs at the airport much less the ticket. Some of these people fall into drugs and other problems and that's not where you want to be in a third world country.
There are great and amazing adventures if you choose to do this but make smart choices. I met a single girl here that spends six months a year taking care of elephants at the conservation center and the other six months in India caring for children with leprosy. She goes back to Switzerland periodically to earn more money. She wears raggedy old clothes and kind of looks like a beggar when you first meet her, but I bet she is leading a really rich and fulfilling life. I dream of making such a difference. But moving away from my 8-5 job in the US is the best thing I ever did and a step in the right direction. I felt like I was dying there, working in the same crap cubicle, dreaming of my next piddly 2-week vacation each year. Going home to watch TV and fall asleep to do it all over again the next day. My life is a thousand times better and just keeps getting better.
Anyway, best of luck to you!!