You can open an account at ScotTrade for free, and then buy a few index funds that you like. You could either put these into mutual funds, like Vanguard's VFINX US S&P fund, or buy exchange-traded funds which track indices as well.
Investing early is wise, the key thing is not to try to beat the markets (at least not with YOUR money on the line!) and also note that volatility will make you unhappy if you check often. You'll be up and down but, over the long term, hopefully up. But if you check frequently, the "down" times will make you unhappier than the happy-times will make you happy. So, set it and forget it, if you take a diversified and relatively low-risk approach.
For 15k, you could invest across bonds, US stocks, international stocks, emerging markets, and more, all without significant cost or overhead -- if you play the broader markets, instead of trying to pick winners.
Most importantly, keep educating yourself, and not just what other people say on random internet forums ;)