Paris¹ is different from the rest of France, including the Paris suburbs².
Outside Paris, you'll find supermarchés (large) and hypermarchés (larger). Depending on what shops you go to in the US, your expectation may be closer to hypermarchés. Hypermarchés are generally located somewhat outside of town, hard to access without a car. Supermarchés are found both in town and outside. A typical grocery shopping experience outside Paris is to drive to a large supermarket and pick up food for a month, and then walk or drive to pick up the odd perishable or forgotten item, possibly from different stores.
Paris has no hypermarchés and relatively few shops that non-Parisians would call supermarchés, but a lot more smaller stores that mostly sell food (supérettes). Due to limited space, most stores only sell a few brands of each product. A typical grocery shopping experience in Paris is to walk to a nearby store once or twice a week (in the big city, who has space to store food for a month?). From Paris, you can drive, bike or even take somewhat inconvenient public transport to reach hypermarchés in the suburbs, and you'll find more choice than in any one Parisian shop (but not more than in the shops within a 10 minute walk of where you live in Paris). Hypermarchés are cheaper, but from Paris, they're hardly worth the hassle.
All major brands of supermarché and hypermarché deliver in large enough towns, including Paris and most suburbs. The prices and selection are not always the same as in store.
One thing that's different between France (both in and outside Paris) and other countries I've been to is that fresh produce in supermarkets is generally inferior and not particularly cheap. Get your fresh fruit and vegetables from the street market (in Paris, there's a street market within walking distance every morning except Monday; in small towns you may be restricted to one or two market days a week). Get your bread from a bakery. Supermarket meat is usually ok.
Outside Paris, most shops are closed on Sunday and Monday. In Paris, most shops are closed on Sunday. All shops are open on Saturday, that's the main shopping day in France. In small towns, most shops close around 18:00 or 19:00. In Paris, food shops including supermarkets and supérettes are typically open until 20:30 or 21:00. Any sufficiently large town has a mom-and-pop neighborhood grocery store with more extended hours around the place somewhere.
¹ I use the local definition of “Paris”, i.e. the Paris municipality, which is basically the part inside the boulevard périphérique, i.e. arrondissements 1 to 20. Other municipalities in the Paris urban area are the Paris suburbs (banlieue).
² Maybe a few other large towns are more similar to Paris. I don't know, I've never lived in a large town in France apart from Paris.