Two things to remember when you are handling introductions: ladies first, and when everyone is the same gender, age conquers all. That means you introduce the older person first.
For example, when introducing a friend to your mother, the introduction should go as follows: "Laura, I'd like you to meet my mother, Mom, meet Laura." The proper response is simply, "Hello, Mrs. Smith," and "Hello Laura, how do you do?" It is not proper etiquette to say nice to meet you.
Women typically do not shake hands in social situations, though it does happen often in professional settings. For men, shaking hands upon introductions is the protocol.
What about being introduced? A couple to tips will see you through any introduction. If you are sitting down in a restaurant or at a party when a friend approaches with someone he or she wants you to meet, there is no need to stand up, unless the person you are meeting is elderly, or a member of the clergy.
As you are introduced, smile and look happy to meet the person you to whom are being introduced. Secondly, really concentrate on learning that person's name. Also, try to have a conversation starter ready to go. Be observant and willing to listen. People who are interested in other people, make for interesting people. In other words, just be considerate because that's really what etiquette is all about.