You can open an online brokerage account at Schwab. It costs $4.95 to purchase or sell a stock. There are also mutual funds available that don't have any fee, but you have to hold them for something like 60 days before you can sell without a penalty.
A few things to remember:
1) never invest what you cannot afford to loose
2) paying off debt can be a guaranteed "return on investment" if you have debt. This is because you save interest you would have paid... and that savings is like a tax free investment. Paying off a 7% interest loan, can be the same as a 10% return on investment, because you have to pay taxes on your investment returns.
3) research - invest in what you understand. If you don't understand an investment, research it until you do.
4) come up with a long term plan, and stick to it to avoid being influenced by fluctuations in the market. But, balance that with having an open mind and being willing to change your long term plan if there's a good reason.
Its fun to see your investments grow. Everyone makes mistakes with investments, learn from them and go on. I've learned a few lessons from not doing enough research in some investments. But its rewarding to see your investment grow when you make good choices.