Hajime mashite: Nice to meet you.
Watashi wa Fany Savina desu. My name is Fany Savina. (lit. I Fany Savina am.)
Yoroshiku onegai shimasu: lit. I'll be in your care / please take good care of me
Yoroshiku onegai shimasu has no actually phrase to represent it in English, but it is roughly another way of saying Nice to meet you, but I placed the literal meaning instead of pleasure to meet you, because saying that twice is strange in any language.
Introducing myself in Japanese