The French shake hands. Even more so than the English or Australians, if you are introduced to someone, or even introduce yourself to someone, you must be ready to shake hands and be unfazed at the prospect.
This applies not only to social situations, but when taking your car in for a service, talking to the builder or plumber, or even walking into a bar. Take your lead from the other person, but be prepared. There is a lot of embarrassment potential in not being prepared to accept a handshake offered in politeness.
I used to shake hands with our roofers every day. It was expected that a cheery "bonjour" be accompanied by a handshake. If I wasn't at home when they arrived (or inside, busy), at the first opportunity we would call bonjour to each other. Quite often this would be accompanied by them climbing down the ladder to shake hands. If their hands were dirty, they would proffer a wrist to be shaken. (I have done this myself if I am working and my hands are dirty (HAH!) and someone calls to visit. Wait until they are within shake range, extend the arm, but with the wrist bent back so they can take the wrist and shake it).
Blokes you would shake wrists with.The wrist shake can also be the proceedure when you are wearing work gloves, but it is better manners to remove the gloves and shake hands properly. If you are both wearing gloves, or both have dirty hands it is polite to wipe your hands on your dungarees and shake the other person's wrist.
There is a variation on the wrist shake, and that is the elbow shake. This is much practiced by car mechanics and plumbers when their hands are dirty beyond the wrist. The proceedure is similar, but with the elbow bent rather than the wrist. I have never seen a wrist to wrist shake, or an elbow to wrist shake. Someone has to use their hands. The only alternative, if you are both excessively dirty, is to look at your hands then at the other person and give a rueful grin. Even then, however, often hands are wiped and some variation of handshake performed.
French men see nothing strange in this. They would see it strange if you offered the Bises though...
(Photograph by Susan's mother.)