1) As a game, in which you don't mind losing all your money. With this method, you'll pick a stock or two based on research, with a solid plan for when to sell it. Potential to lose everything or multiply your money by 10 in a year. Guess which is more likely.
2) As an investment, in which you buy a total stock market index mutual fund (Vanguard is good), which gives you a share of essentially the entire U.S. stock market. As the market fares, so do you.
There's actually a third option, which is research and put together a diverse portfolio of your own stock picks. The fact that your asking the question means you have neither the time, money, or knowledge to pull this off well. (Don't feel bad - very few people do.)
Of course, stock purchases should be only one part of a comprehensive financial plan. In basic order of priority, you want to 1) Pay off credit card debt, if any. 2) Establish a 3-6 month emergency fund in a liquid savings account. 3) Max out any employer-based retirement investing options you have (especially with any employer matches). 4) Save money on a regular basis, a fixed amount each month. This money should be allocated according to when you'll need it. For retirement, mostly stocks, for anything less than two years away, mostly money markets.
I'm trying to think of the best book to reccomend, but nothing springs to mind.
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Also, if you're actually planning on making money, you probably fall into one of two categories.
1) You think of it like a good place to store your money and let it accumulate interest, only faster than a bank.
If this is the case then take Dag's suggestion and buy one of the indexs, Dow for example, or a really large, diverse, mutual fund. The stock market may go up and down over even a few year period, but if you can sit on your money long enough, your pretty much garunteed a profit, above bank interest as well. But you have to be able to think and invest long term this way.
2) You want a lot more money, and you want it soon. This is kind of day trading policy. If you really need the money I don't really suggest doing it this way, but, well if you're going to here's what I suggest. Do some serious reasearch, best if done in an area your already framilar in. Look at the companies and see if there's one that you think is clearly undervalued is will sky-rocket soon, either due to some new thing in it's pipeline, or something overlooked that will come out soon. Be ready to pull out your money either when you think it's peaked (have to figger that one out on your own, if I knew how to tell when something peaked I'd be a lot richer ) or if the stock doesn't jump the way you think it will.
If you're not trying to make money but just want to have fun, first figure out how much money you're willing to loose, and then organize your finances as if you no longer have that money. Then invest however you want. Everyone has a prediliction for different types, long-term, short-term, in between, whatever you want, in this case your in it for the fun, not the money.