I have been searching the net for a really good image of a French infantryman of 1870 - partly because it is a period that I like and I want you to like it too...OK? This is one I chose from Louis Delperier's article on French Infantry of 1870 in No 12 Military Illustrated Past and Present. Delperier seems to be the most knowledgeable author on the subject and has written many articles on the period for Tradition Magazine.
The uniform is one that is based on the campaign experience of the French army in North Africa - the kepi copied across the world's armies had its origin in a peaked version of the fez and the greatcoat being the prinicipal garment was another modification from North Africa. The epaulettes got left off in time - this reconstruction is apparently from August 1870.
How I would go about reenacting this period is something that has been on my mind for a few days and here is my thoughts on the subject. I reckon the period of the Third Republic is most famous for its art and culture - so a group ideally would recreate the whole Belle Epoque - cycling, Impressionist painters and radical politicians could run shoulders with the military bit. The period up to 1914 could be done at Heritage railways and boating lakes, picnics and the infantry wing could be ready for the centenary of the Great War in 2014.
I think there is a group doing this in France but it could work anywhere - the US, UK anywhere...just my thoughts on the subject. Feel free to comment...