You may start from your neighborhood computer institute. Ensure they have the proper soft and hard infrastructures. Another important thing is their software licenses; please ensure that the institute is not using any pirated software.
Another way to start your part time software trainer career is to start your own home institute with freeware. Most of the licensed softwares are costly so if you are an expert in some freeware it will be easy for you to start your own home institute.
There is one more way: go online, find some site like liveperson.com or guru.com, etc. and start from there.
Software trainer is an exciting part time career option for all who deal with any software in a day to day career and who has a passion for teaching. Some of the technologies that use part time software trainers are: Microsoft technologies, java, C,C++, pro engineer, catia, ansys, AutoCAD etc. Finding a software institute nearby your house is the best option to start other wise you can start your own institute or go online.