French opening and closing hours:
To give you the bottom line first, it varies, and one needs to carry extra provisions in the event.
Generally, shops and restaurants will be closed on Sundays and one additional day of the week (could be Monday, Tuesday, or Wednesday). In the very small towns (ie, the ones you are likely to be in), that means everything closes. In slightly larger (two-restaurant) towns, sometimes place A closes on Monday and place B closes on Tuesday. But one can be hungry even in the large towns: I arrived in Cahors (major town - 50,000+) on a Sunday evening to find every single restaurant closed. Frankly, I was convinced that whenever they saw me coming, the town closed down!
In France, everyone goes home (or out) for lunch, so shops close from about 12 to about 2. Everywhere. Restaurants are open for lunch from about 12 to about 2 or 3, and then close until they open for the evening meal about 7 or 7:30.
Bakeries will often be open early, and for a few hours, on days others will be closed (Sundays and "closing day" whenever it is), where you can get sandwiches and the occasional pizza slice. However, there is enough inconsistency in this practice that there were continuing sources of disappointment.
Your best bet is to reserve your lodgings ahead for Sundays and Mondays; either get demi-pension or confirm with your host that a local restaurant will be open. If they are self-catering, make sure you stock up groceries on Saturday. Also plan your lunch supplies for Sunday, Monday, and Tuesday.