People in the United States don't always shake hands when they are introduced to one another. However, in a formal or business situation people almost always shake hands.
1.A: Mary, this is Joe's brother David. B; I'm very glad to meet you. C: It's a pleasure to meet you. B: How do you like Texas so far? C: It's really different from what I expected. B: Don't worry. You'll get used to it in no time.
2.A: Mrs. Smith, I'd like to introduce a friend of mine, Pierre Dubois. B: How do you do? C: Hello. B: What's your impression of the United States? C: Well, I can't get over how different the weather is here. B: Oh, you'll get used to it soon!
3.A: Wendy, I'd like you to meet my brother Sam. B: Hi. C: Nice to meet you. B: What do you think of Dallas? C: Well, I'm still feeling a little homesick and so many things seem strange to me. B: You're bound to feel that way at first, I guess.
4.A: Mrs.Hughs, this is Peter Brown. B: Pleased to meet you. C: How do you do? B: I hope you're enjoying your stay here. C: If it weren't for the climate, I'd like it here very much. B: It always takes time to get used to a new place.