That is pretty sad. When my sister used to babysit about 7 years ago she'd make about $8-10/hr.
Stocks are 95% beneficial long term, unless you invest in a small company that goes multi-million in a few months, you'll be out of luck for immediate return (better chance at winning the lottery). It's best to invest a lot right off the bat in order to get a decent ROI. Investing is never a bad thing though, the market is constantly going up (in the long run); take a look at mutual funds to begin with, then as you make more money, start thinking about stocks.
As for a quick $, maybe ask the people you babysit for if they will pay you extra if you wash windows/vacuum/dust/etc. while you babysit. You can also look into dog walking. Also, ask you mother to ask her friends if they could use your help.