I really like JKS's suggestion of the community college.
Personally I have never done any MA teaching. However, for a living I do systems engineering which is kind of like high-level tech support. Part of my day is spent solving problems, the other part is spent explaining stuff.
Take any opportunity you have to teach whatever you can...including material outside the MA world. I noticed on another thread that you are a musician. Try your hand at teaching music, either to a friend, or in the community, or even as a job if you have the background.
Check a local library or other centers in the community (church, YMCA, parks department, etc) to see if they have any volunteer teaching jobs. Many volunteer opportunities, such as Adult Literacy or English as a Second Language can result in some basic training as well as materials, all you have to bring is desire, willingness, and a bit of patience.
Step forward at work and be the one that volunteers to train up a new employee and get them ready for life on the job. If you are a college student, volunteer to give tours for prospective students or help out at New Student Orientation. If you have other skills -- for example, if you are good at graphic design, or if you're the company computer guy -- see if you can volunteer teaching these in the community as well.
Something that I have found is that what you learn along the way can apply to all aspects of teaching, training, or even just explaining. A specific example: I became a lot better at working with people in other nations after volunteering as an ESL teacher and working with people that have heavy accents and speak broken English. A general example: attempting various teaching and training tasks has taught me how people learn and perceive material in different ways, and that an example that may work for one student may not work as well for another student.
Teaching/training is a challenge, and a big responsibility. You are the face and the voice of what you are teaching. You don't have to be perfect, but you do need to do all you can to represent it well. You'll find things you don't know. Don't BS, tell the student that you need to get more background -- then (most important) be sure to follow up with that student promptly. You'll learn a lot. It will also tax you, mentally. Be prepared for that and don't let it stop you. You will get students that don't want to learn, and that may frustrate you...but you will also get the students that love how you have opened doors for them, and that will (hopefully) make it all worth it.
