No doubt you are aware the stock market has been a trifle unpredictable lately. Crude is down, then it's up, and the whole market goes with it. Like real estate, the stock market is cyclical. "Buy low, sell high" is an ancient axiom. Right now there's a strong possibility you can at least take that first step. There are currently a lot of relatively bargain-priced choices, so it's a good time to try investing on your own.
Decide how much you are going to risk and accept on faith that you don't want to put all your money into one stock. Take a diversified approach, putting a little bit in several different stocks.
Set up a brokerage account. ETrade, Fidelity Investments, Scottrade, Ameritrade, and Charles Schwab all handle these types of accounts. Free to set up, you're required to deposit money to use them. Minimum amounts vary.
Research. Do the research. Don't take somebody's word for it when you can look it up for yourself. And you can look up almost anything online. Look at the stock's history, its current trend, how much you would have made (or lost) on it if you'd invested six months ago.
Think about reducing your risk by investing in funds. There are two types, Exchange Traded Funds (ETFs) and mutual funds. The latter are groups of individual stocks that are traditionally considered safer investments. Because mutual funds can trade only at the close of the market day, you're in the muck if something b-a-d occurs during the hours beforehand.
Take it easy. Start off slowly. Do your buying over several days or even weeks. Keep an eye on your investments. This is your new job. If one of your stocks goes down a little, buy some more. If it goes down a lot, recognize you're being given a sign. Generally, the correct interpretation is that you should get out pronto in order to limit your losses. The stock market is a place where you can make a lot or lose a lot far quicker than you would have ever dreamed possible.
Because the market goes in cycles, be prepared to ride it out. If you have a largish amount of investment money, you should probably hire an investment advisor. They get paid a percentage of the amount of money managed, so it's in their interest to watch the market closely all day and make your financial future bright enough to shine on both of you.