"Onegai shimasu" お願いしま is a hard phrase to directly translate to English. The second part "shimasu" is basically the verb "suru" which means "to do" conjugated into the present tense. "Onegai" comes from the verb "negau" which literally means "to pray to (something)" or "to wish for (something)." The "O" at the beginning is the "honorific O" that makes the phrase more "honoring." Of course, we would never say that particular phrase without it, but that's what it is. (Don't confuse this "O" with the "O" in O-sensei. The one in O-sensei is actually "Oo" meaning "big" or "great.")
In Japanese culture, we use "onegai shimasu" in many different situations. The basic connotation is the feeling of exchanging "good will" towards the "future" of the two meeting parties. Hence, it's sometimes kind of like saying "I'm hoping that our relationship holds good things in the future." We use this during New Year's celebration by saying "kotoshi mo yoroshiku onegai shimasu" which transliterated (to the best of my abilities) means "this year also good tidings I pray I do." You get the gist.
Another connotation is "please" as in, "please let me train with you." It's an entreaty often used in asking the other person to teach you, and that you are ready to accept the other person's teaching.
If you're feeling really, really humble, you can say "onegai itashimasu" which uses "kenjyougo" or the "humble" form of the verb. This places you lower on the hierarchy than the person to whom you're speaking (unless they too use the same humbling form -- in which case, I think it all comes down to stuff like who can put their nose closer to the ground when bowing or something).
To pronounce this, I'd transcribe it as:
o ne gai shi ma su.
(If you want to get technical about it, the last "su" is a stop-fricative rather than a fricative-vowel combination, so it _is_ pronounced like the ending "s" in "gas" moreso than the long "su" sound in the name "Sue.")
