starting off with 300 dollars of credit, my payments are only $15 a month.. When they up my credit limit to 500, would payment increase as well?

It depends on what type of "plan" you have. A basic credit card will let you buy things with a 30 day interest free period, meaning if you pay off what you bought by the due date of your bill, you pay no interest. But, if your credit card gets maxed out (you use the whole limit) and you don't pay it off completely, then YES your bill will be higher. It's based on the amount of credit you have spent.

Your payments are based on your balance, not your credit limit. So if you don't use the extra credit, there would be no increase in your monthly payment.