I want to get into running this game but I can't find anywhere (even elsewhere on the internet) what the requirements are for unlocking each stage. I figure this should be basic information available in a guide, was wondering if anyone knew off the top of their heads and could tell me.

Sorry for taking so long to reply! To unlock the stages you just need to collect a specific number of gems. All you need to do is clear your save game from the game menu, then when you pick a character you will see each stage is locked except the first stage. It will say 0/4 or 0/6 or 0/12 or whatever on each stage. That is the number of gems you need to collect on that stage in order to unlock the following one. So you can get all the exact numbers in the game by clearing your data and going to the stage select.

Gotcha, thanks! Don't know why I didn't think to do that... here's my next question, what makes a gem spawn exactly?