1. Study the company you want to trade
Don't buy stocks only because some numbers told you to do so. Be sure to understand the difference between market capitalization and the value of the company itself. Please read the recent annual and quarterly reports of the company (SEC 10-Ks and 10-Qs). Read all possible analyst reports and make your judgement.
2. Study all legal and tax requirements
Trading stocks is not an easy thing for a beginner. You have to be sure that everything you make is legal and won't harm you in the future. In some countries if you sell a stock back in less than six months you would have to pay additional taxes apart from the fees to the trading system. Check if you can buy international stocks or if it is better for you to stick to your local ones.
3. Understand your trading system
If you use an online broker, always compare the fee against its competitors. You have to know all possible costs before you send them an initial deposit of funds. This initial deposit can be from USD 500 to 2000, so you're not playing with a small money you can easily loose. Look for all full-service brokers, discount brokers and normal online brokers and find one that suits you the best.
If you're depositing in other currency than USD, please bear in mind possible fees or other costs associated with exchanging the currency to USD in which majority of blue chip stocks is traded. Try to estimate if all transactional fees are not higher than your predicted gain from the stock. If you pay USD 10 for a transaction, but after selling a stock you get just USD 5 of additional money, then this is probably not worth the hassle.
4. Start slowly
You don't have to buy all stocks that you consider to be blue chips. It's not a shame to buy just few stocks a month and don't trade for several weeks or months.