There are guides about the etiquette of introductions. The rule of thumb, according to Emily Post, is to introduce the youngest to the eldest, the most junior (sales consultant) position to the senior position (president of a company) and the business person to the official (mayor or politician). Very few people are aware of the importance of business etiquette, yet good etiquette gets noticed. This offers you one more tool to make a good impression.
Practice your introductions until they become very smooth. At a mixer in Oakland, a man tried to introduce me to the mayor, Jerry Brown. The man was obviously uncomfortable and fumbled through it so poorly that we all felt uncomfortable.
According to Emily Post, if a woman doesn’t extend her hand, the man shouldn’t either. A proper handshake is when both bodies are squared off, not halfway turned, which is a defensive stance. Keep an arm’s length between you. When clasping hands, both thumbs should be parallel to the floor with the web between the thumb and index finger touching on both hands. A count of three is sufficient. Briefly look into their eyes to show a genuine interest.