Hurray! You've met another Japanese-speaking person. Time to introduce yourself.
But how do you say "Pleased to meet you" in Japanese?
The first phrase you'll want is:
"Nice to meet you"
Hajimemashite (almost literally) means "we are meeting for the first time". So you can only use it the first time you meet someone.
The other super-useful phrase is:
Yoroshiku onegai shimasu.
"Please be kind to me."
Yoroshiku onegaishimasu is hard to translate, but means something like "please be kind to me".
It means that you are looking forward to having a good relationship with someone.
Make it more polite
Add douzo to make your greeting more polite:
Douzo yoroshiku onegai shimasu.
"Nice to meet you" (polite & a bit formal)
You could also say:
O-ai dekite ureshii desu.
"I'm happy to meet you." (more polite & formal)
O-ai dekite kouei desu.
"I'm honoured to meet you." (even more polite & formal)
Keep it casual
If you don't feel like being so polite, you could also say:
"Nice to meet you" (a bit more casual)
"Nice to meet you" (very casual)
It's good to be nice-mannered when you meet new people though, right?
"Nice to meet you too!"
Last but not least, when someone says yoroshiku onegaishimasu, you can add the feeling of "me too!" by replying with kochira koso ("me too!"):
Kochira koso yoroshiku onegaishimasu.
"No, I'm pleased to meet you." / "The pleasure is mine."
Now, go and find someone new to speak to, and tell them how pleased you are to meet them.