Useful Greetings for early meetings

When you first meet someone,

A: Nice to meet you! (I’m) (your name).

It’s a pleasure! (I’m) (your name).

or Hi. (I'm) (your name).

or (your name)

B : Nice to meet you, too. I'm (your name).

or The pleasure is mine. (Your name).

Ex.

Jerry: Nice to meet you! I’m Jerry

Elaine: Nice to you meet you, too. Elaine.

Ginger Rogers: It’s a pleasure to meet you! I’m Ginger!

Fred Astaire: The pleasure is mine! Fred!

If you haven’t met someone, but are standing near them, it is ok to introduce yourself. A common expression for this is

A: I don’t believe we’ve been introduced. I’m Tom Hanks!

B: Nice to meet you! I’m Debra Winger.

Note:
Though most language books teach people to say “Hi, my name’s (your name)”, this is a practice that is really only done by children among native English speakers. Though in many cultures, just saying your name when you meet someone may be seen as rude, this is a very common practice in North America.

By chance, you see a person in front of the bank on Spring Garden Road.

John: Excuse me, haven't we met (before)? You look familiar.

Uma: Good to see you again. You're John, right?

John: Yeah. So, how are you?

Uma: Couldn't be better, and you?

John: I'm great. You must be waiting for someone.

Uma: Right. One of my co-workers is having a housewarming party tonight. So, I'm here to meet another co-worker to go with.

John: Oh, my friend is here. I've got to go. Nice talking to you!

Uma: To you, too. See you around.*

*’See you around’ is a common expression to say, “I’ll see you around town” or “I’ll see you around work”. It’s a casual expression usually used between two people who do have a good chance of seeing each other again.

You meet a friend in a night class

Salma: Hi, Antonio. What a coincidence !

Antonio: Yeah, what a small world! Are you taking this course?

Salma: Of course. Looks like you are, too?

Antonio: Yeah, I guess it'll be more fun studying with someone I know.

Salma: I agree.