Hi, Edgar :) You can say "Hajimemashite" when you meet someone for the first time. "Douzo yoroshiku (onegaishimasu)" means "please look after me from now on" as ryan said, but also you can say it when you ask someone to do something instead of you. Like when you ask someone to send email or show you textbook or something, you can say it. Also "Douzo yoroshiku onegaishimasu" is the most polite, but you can say just "Yoroshiku" to your friends!

Hajimemashita is the first time meeting, if I remember correctly douzo yoroshiku is a more formal way of meeting someone land also means look after me from now on, its been a long time since I've looked up the full meaning for these but I think its right