A few points. None of them particularly encouraging I am afraid. But best to know what you're getting yourself into.
Teaching schools do not want travellers. They want you to stay a year or more in general. Two and three month contracts are around yes, but they are designed at the convenience of the school ("flexibility" it's called in UK) not the workers. Generally speaking, you need to make a longish commitment of time to the job. I am sure there are exceptions, but this is generally the case. Look at it from the school's point of view.
Fulfilling Simon? No. this is not the reason to do TEFL. You are teaching businessmen, and disinterested well-to-do teens whose fathers want them to acquire the status English-speaking acrues. It is unlikely that you will get work with either children or dedicated learners.
And yes, the pay and conditions are lousy. Enough to live in the country. But not to save and travel. If you want the experience of living abroad, and are prepared to live at a low standard then TEFL is a good option. It suits early twenty-somethings I reckon. A life of coffee and scruffy trousers.
Exception: Japan. But here you are something close to a full-time professional teacher, and have proper responsibilities. It is not really for the "give it a try" types.
China will take anyone at the moment, but, again, the pay is rubbish and you are expected to live for a year in the middle of nowhere. Is this your life-plan?
Do you like teaching? Have you ever done it? Really, it doesn't suit everyone. I find most people have a completely warped notion of what the job involved. Funny, given everyone has the experience of schooling. So talk to some teachers.
This is a very funny site, written by a TEFL teacher:
http://chasemeladies.blogspot.com/
Scroll down for his teaching entries.
Lastly, to be more positive, you live in a country with a strong currency compared to most. Your best option for money is on your doorstep.
Here's an idea. Just for spice. How about six months touring India? Fly there. Buy a bike there, and resell at the end. Monthly expenses? Upwards from 350GBP. 450 is comfortable. Total: three grand will get you a fine experience. If you fancy SE Asia instead, then make it four or five. It is a lot, but not an impossible amount to save over a year or so. Maybe you own some assets, like a house and a car? Get a smaller ones.
I'm not trying to put you off here - just share my experience as a teacher and traveller. Biking travel is great. But TEFL is suited to a particular type of life. A stationary one. If you want an in-depth experience of another place then it may be for you. But it fits poorly with the hair-in-the-wind, whereever-I-lay-my-ignitions-keys-that's-my-home life that most people have in mind for biking travel.
Hope this helps,
Simon