(Under construction)
Researching the history of local government will, of course, depend in part upon the country in question.
What material is there already available on local government in the country? This can include official websites, Wikipedia, published books and any other sources, including archives and newspapers/magazines etc.
Get a list of current local government units and a timeline of developments - legislation etc.
It may be appropriate to create a wiki or other system to organise information - not all of which will be gained sequentially.
Define the body being researched, its geographical extent, timeframe and other aspects.