Adding to tx5smith's answer, if you save your money in a certified Credit Union, they will pay you interest. It would be a small amount, but you wouldn't have the risk of losing it on an investment.
If you decide to invest, make sure you only risk an amount you are willing to lose...because the nature of the risk (in most investment forms), means that someone has to lose money for somebody else to gain money. So, say goodbye to invested money and if you are very fortunate, you will get it back.
Saving $300 a month for 5 years would give you $18,000. If you invest the money every month you would have to factor in the interest or return on the investment and when that return is paid. You can't know the actual amount you would have at the end of 5 years unless you have all of the fees, interest etc.