Ouch!
To start with how old are the kids? Will it be just yourself and the kids? Most important, do you speak any language other than (American) English? What sort of holiday do you want? (seaside, country, places of historical interest etc?)
France is a great place to visit – if you speak French. In many places you’ll find the locals will simply shrug their shoulders and turn away if you can’t, and the places where you will find English spoken by local people will generally be the Tourist Traps.
How much are you expecting to pay? Some places cost a fortune, for example to rent a mobile home (trailer) on a Holiday park, NOT the same as a US Trailer Park, you will be looking at around $800 – $1000 a WEEK in the high season at somewhere popular such as Frejus or St. Raphael, by NO means the most popular holiday destinations on the Cote d’Azure. (South of France)
Any food hang up’s, allergies, problems? That can also be an issue because there’s only so many days anyone can live off Big Mac’s.
House exchanges are unusual to the point of being virtually unknown in most of Europe. Holiday rentals are expensive in the popular (that means Tourist) spots and where they are inexpensive usually means inaccessible and remote.
There is one point you make that I would comment on, and don’t take this wrongly, it’s not intended to be hurtful. You write that you would “Like to get to know people”
Firstly you make the assumption that people would like to get to know you.
Although in the US people are inclined to be ready to establish friendship with others, especially foreigners, in Europe such is not the general case. In fact American people are not well received in many parts of Europe these days.
Tolerated, mostly yes.
Welcomed? That’s another matter.
You’re unlikely to meet open hostility (unless you stray into places such as the French “banlieues”, places often confused with being a French version of US inner city slums which they are NOT) and where if you opened your mouth in the wrong place at the wrong time life would turn very nasty very quickly.