The custom probably comes from medieval times when people took off their armoured gloves when meeting and shook hands. A recent study published in the Journal of Psychology, suggested that a firm handshake makes a good first impression. They defined a firm handshake as strong and vigorous with a complete grip that did not stop too soon or go on too long. Add eye contact and the hint of a smile and everyone is impressed. The research was probably done by a man.
Women seem to have slightly different rules.
Try these rules - Extend your hand first and leave it there until the other person responds. Some people are reluctant to shake hands but will get the idea eventually. If their religion forbids contact, you will get the message and you can sheepishly put your hand away. Push until you contact the fork between the thumb and the first finger. Don't get caught half way in and don't squeeze the ends of people's fingers.