Hajimemashite: How do you do?
Doozo Yoroshiku: It's nice to meet you.
A simple introduction, when meeting a person for the first time, could go like this:
Person 1: Hajimemashite. Rondillo desu.
Person 2: Hajimemashite. Lee Hyori desu.
Person 1: Doozo yoroshiku.
Person 2: Doozo yoroshiku.
Translation
Person 1: How do you do? I am Rondillo.
Person 2: How do you do? I am Lee Hyori.
Person 1: It is nice to Meet you.
Person 2: It is nice to Meet you.
You could also add a few of these greetings with the dialogue:
O-namae wa: What is your name?
Ohayoo gozaimasu: Good morning (formal).
Ohayoo: Good morning (informal).
Konnichi wa: Good afternoon.
Konban wa: Good evening.
O-genki desu ka?: How are you?
Shitsuree shimasu: Excuse me (as in good bye).
Ja mata: See you later.
Ja: See you later (informal).
Sayonara: Goodbye.
O-yasuminasai: Goodnight.
for more expressions, go to this
thread.