There are basically two ways of driving through france. Through the "route national (RN)", and the "Autoroutes". The autoroutes are the quickest and most convenient network in France. However, if you have the time and want to enjoy the beauty of the countryside the route national provides a valuable alternative for driving through France. Another advantage of the RN is that they are toll-free, whereas most highways have tolls.
As @Gilles already said the highways have regular rest areas, they are called "Aires de repos". There is a website where you can find the offered services per rest area. This website is in French, and also not handy since it uses the French numbering (Axx) to search, whereas the Google Map used, uses the European numbering (Exx)
In general you have two types of "aires de repos". The first being offering services like restaurents, service stations, sometimes a hotel, etc. The second being the basic version with at most only lavatories.
You can distinguish the two by the panels indicating the aires on the highways:
The ones with extended services:
(Source: Google maps)
and those with basic services (typically containing an icon with a fir tree):
(Source: Google maps)
Personally, I would not recommend staying the night at basic stops (the ones with the fir-tree icon). The ones with services can provide a relatively safe stop to spend the night.
Regarding the showering, some do provide showers, but it is not common. Your best chances are where the truckers rest as well.
On the RN, the aires de repos with services are not that common. Usually you'll find the services in the cities the RN connects. Just before entering a city check for the budget hotels. F1, Camponille, and Ibis Budget are examples of budget hotels throughout france where you can stay and take a shower. They provide a very basic formula catering to drivers. They are open 24 hours a day.
I have spent the night in my car while driving on a route national, but it is something you need to decide for yourself whether you are comfortable doing. I would not recommend staying at a supermarket car park at the city boundaries. Local bored youth might disturb your night's sleep. What I do is search for a nice spot in the country side, park my car out of sight and take a nap.
You can also buy or bring a tent and stay at the various campsites.