Kyrgyzstan's greatest export is its people departing for Russia, Kazakhstan, and even Europe for better opportunities. There are few opportunities for foreigners, except with development organizations, that generally hire off-shore. There are also few opportunities to teach European languages, as many Kyrgyz that studied abroad have returned with near fluency and will charge much less than you.
If you wish to volunteer, there is a very active and diverse NGO community that would appreciate your assistance.
If you come to Kyrgyzstan on a 30 day tourist visa, you will not be able to extend your visa. If you come to work or volunteer you should apply for a longer term visa. Also, Americans and Europeans on long term visas must register with OVIR. Information on this process is available on the websites of the US Embassy. It is an easy, inexpensive, straight-forward process not fraught with corruption or delays in most cases.