Elude, first of all, have a good time.
Staying off the Interstates is good advice. Use them only when you need to get from A to B as rapidly as possible. They are exactly the same as Motorways in the UK and whatever-they-are-called in France.
Do not underestimate the size of the U.S. Make sure you have an accurate underdtanding of the miles between where you are and where you want to be.
Much of the territory you are interested in is rural, some very, very rural. Don't count on being able to find decent overnight accomodations in the little town you just happened to drive into in the early evening. My recommendation: Book one day ahead. That is, each evening, decide where you want to stay the next night and get on the phone or the laptop and make reservations. (But, don't count on Wifi or another internet connection being available. Rent a U.S. cellphone with coverage for the area and carry maps and hotel/motel directories.)
B&B's exist in the U.S. -- not really noted for their economy -- but few independent or family-managed hotels exist. Almost certainly, you will stay at instances of chain hotels and motels, typically located at Interstate exchanges, in resort areas, or in the downtown areas of cities.
If it was me, I'd focus on the coastal areas of Lousiana, Mississippi and Alabama, then the mountains of east Tennessee and North Carolina (check out Asheville on the web) and the farm and horse country of central Kentucky around Lexington. Do go to at least one horse farm.
Meeting people: Be an extravert. Open your mouth and talk. People will talk back. Very likely, they will be surprised and happy to see a French tourist in their town. Buy the local newspapers in the little towns you visit. (They be in racks along the street, etc.) See what's going on. Expect to see lots of activities centered on local schools and clubs. In August, you will be in prime high school and amatuer baseball season.
Speaking of baseball, minor league teams will be in several of the cities on or near your itinerary. The major league Braves are in Atlanta. A very good minor league team is in Louisville, and Cincinnati in Ohio is home to the Reds, the oldest major league team in the country. (Cincinnati is about two or three hours north of Lexington, Kentucky, on the northern shore of the Ohio River.) Game tickets should be available at the gate except in the most unusual of circumstances.)
Where to eat: Many, many franchise and chain places are located pretty much everywhere. If you want to eat elsewhere, you may need to do a little research. Independent restaurants simply do not exist in many small American cities.
Do not be put off by stories of alleged American disdain towards the French. Most Americans have never met someone from France. We're really a pretty friendly bunch of folks.