It's not that hard...
No, Shimano groupsets do not fit the same frames. You'll need a mountainbike groupset,
something from this range.
Cassettes don't fit on rims, they fit onto the rear hub. Without knowing what hubs you have, it's impossible to tell you what to fit.
A quick Google suggests that a Quest Omega isn't a particularly high end bike, so don't go throwing $$$ at it. Is there a local bike shop who might be able to help?