NB1 & NB2: Introductions

When you meet someone the first time, it is common to greet the person with "How do you do?" the correct response is "How do you do." Here is a short introductory conversation:

Ken: Peter, I'd like you to meet Mary.
Peter: How do you do?
Mary: How do you do.
Ken: Mary works for ...

A variation is also "It's a pleasure to meet you." or "Pleased to meet you."

Ken: Peter, I'd like you to meet Mary.
Peter: It's a pleasure to meet you.
Mary: How do you do.
Ken: Mary works for ...

Ken: Peter, I'd like you to meet Mary.
Peter: How do you do?
Mary: Pleased to meet you.
Ken: Mary works for ...

In informal situations, especially in North America, introductions are also made simply saying: "This is (name)." It is also common to just say "Hi" or "Hello" as a response in this informal setting.

Ken: Peter, this is Mary.
Peter: How do you do?
Mary: Pleased to meet you.
Ken: Mary works for ...

It is also quite common to shake hands when you are introduced. After the initial introduction, hand shaking generally takes place in more formal, business situations. Otherwise, people just say "Hi."

NB1& NB2: Saying goodbye

Here are a number of phrases used when saying goodbye to friends or family as they depart on trips, both short and long.

Long Trips, Vacations, etc. American English
Have a good trip.
Enjoy your vacation.
Have a good time in (destination)

Long Trips, Vacations, etc. British English
Have a good journey.
Enjoy your holidays.
Have a good time in (destination)

Short Outings
Enjoy! (American English)
Have a good time at (destination place such as a restaurant)
Have a good time in (destination city)

When your friends or family return home it is common to use one of the following phrases:
How was your vacation? (holiday in American English)
Did you enjoy your time in (destination)
How was your journey / flight / trip?