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Each of the online brokers (Scottrade, TD Ameritrade, Fidelity, etc) provide everything you need to buy and sell stocks through them.  They are the vehicle that you will use, once you know what stock it is you want to buy.  One major item that used to separate them was the price you had to pay to buy and sell a stock through them, but now even that has pretty much evened out.  As I write this, Scottrade is $7 a trade which I believe is the lowest by a dollar or two.  Once you select the online broker that you will use, you simply need to log on to the site and set up your account.  Once it is funded, you are ready to start trading.

Like with most things, there is a lot of background noise around the actions that one actually needs to know when buying stocks.  Although there are several different types of orders to use when buying or selling stocks, most people only need to know one.  As long as they are not buying tiny companies, a market order is the only order they need to know.  A market order says I want this many shares of this stock right now at the current price.  If the company you are buying stock of has a daily average volume of at least 300,000 shares, and you are buying less than 1,000 shares, a market order will get you very close to the current price.  Once you recognize this, you only need to figure how many shares of what company to buy.