I've been living and teaching in China for about 30 months now. There are some good opportunities to be found in China, but you really have to approach coming here with your eyes open.
"I was just wondering if anybody could recommend an organisation that 'supplies' teachers for learning English in another country."
Beware of recruiters who supply teachers for a fee. No doubt there are some honest recruiters, but most of the horror stories I've heard are from people who have been recruited by unscrupulous agents and end up in some pretty desperate situations in terms of housing and work conditions.
Wherever possible, deal directly with the school which will pay your salary. Ironically, the staff and owners of many English schools don't speak English. This is one reason recruiters flourish. Always ask to talk to a teacher currently working at the school.
I like working in China, and I wouldn't discourage anyone from coming here, but you've got to do your homework. I recommend that you visit a forum that deals with helping teachers to avoid the pitfalls of working in China. Spend some time reading FAQ's and archived posts on this site, and you'll be better prepared to assess job postings.
This is a free membership site that is run by an experienced teacher working and living in China. They do not accept job postings from known recruiters. These are the good guys.
Raoul's China Saloon (V3.0) - Index
Many jobs here require an undergraduate degree or better, but, generally speaking, there's a position for everyone. Demand tends to outstrip supply, so concessions can be made. Have nothing to do, however, with anyone who will not obtain a proper work permit for you.
Do your homework!