I don't know what you mean by " We have just realized that France does not have a point to point public transport system. " France has an excellent rail system and buses too but you need to understand that small charming towns are not well served by public transport. Some regions have better public transport to places of touristic interest than other regions. Around Colmar there is good public transport to places of interest. Provence is a large region and you don't mention where you'll be going in Provence but there are many places of interest served by public transport, specifically the coastal locations. The Loire region has a good public transport infrastructure as well. A good website to learn about rail travel in Europe is www.seat61.com. Here is some info about getting around France using public transport:
There are three types of websites you'll need to get familiar with using: 1) train websites 2) bus websites 3) tourist office websites.
Look at this link to see a map of where the regional and long distance trains go in France:
http://www.bueker.net/trainspotting/map.php?file=maps/french-network/french-network.gif
Here is a map of TGV train lines in France:
http://medias.sncf.com/sncfcom/pdf/destinations/Carte_ReseauTGV.pdf
Here is a map of Intercités train lines in France:
http://www.developpement-durable.gouv.fr/IMG/pdf/Carte_TETallegee-20140227.pdf
To see maps of the local train routes for each region go to www.ter-sncf.com and look at the map of France on the right side of the page. Hover your mouse over the region that interests you and click on it. Scroll down to the bottom of the page. Each region will set up its page a little differently but under the various topics and headings at the bottom look for the words, or combinations of words, that include “carte réseau” or “carte” or “réseau”. Sometimes you won't see any of these words and in that case try looking under the topics in “Services & Gares”. For example, in the Provence region you look under “Services & Gares” and click the topic “a votre service”. Then scroll down the page and you'll see “Carte du Réseau TER PACA”.
You can use the TER website to search train schedules and prices in each region but it is in French only. You can also use the SNCF website to search for schedules and prices. Keep in mind that for any train journey that requires a TGV train and most Intercités trains you will get the cheapest prices by booking three months in advance. Last minute fares for these trains will be the most expensive. The TER trains are local trains and have fixed prices and no reservations are needed.
To buy train tickets and check schedules and prices for anyplace in France you can use www.capitainetrain.com. Remember, buy tickets up to 3 months in advance for TGV and Intercités trains. You can also use www.voyages-sncf.com for the same purpose but it has a tendency to redirect people to the RailEurope site, which should be avoided as it has higher prices and doesn't show all available trains.
Look at this link for bus info. It has a list of all the départements in France so find the département that interests you and click on the link and you'll be directed to the bus website for that département.
http://www.itransports.fr/fr/reseau/departement
You also need to learn how to use tourist office websites. You should google the tourist office websites for any region, département, city, town or village you may want to visit. You will find loads of info on these websites including hotel/accommodation and restaurant info as well as what to see and do in the area. Occasionally the websites have English versions. In doing a google search enter the words "office de tourisme" followed by the name of your region, département, city, town or village and this will bring that place to the top of your search.