I agree with Kylecito. I'd like to add that "Yoroshiku onegai shimasu" is used not only when you first meet someone also, you use after you request someone something to do. e.g. この問題を解決してください。（Please solve this problem). よろしくお願いします。　（Please (for emphasis, in a polite way)). You can never say "Hajimemashite" in this context. Maybe "How do you do" is the closest translation for "Hajimemashite." I agree with the translation of "Yoroshikuonegaishimasu with Kylecito's".

As far as i know, 'hajimemashite' means literally 'let us begin', and is used when meeting someone for the first time. 'yoroshiku onegaishimasu' means something along the lines of 'please be good with me', and is used after 'hajimemashite'. I've seen it also used as a greeting, especially on the internet as 'yoro' or 'yoroshiku', but i don't know if this is actually right