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It's pretty simple to buy shares, you sign up to a decent broker. Make sure you select the correct share, i.e. know your stock code. Know your costs; to buy shares you pay for the shares plus 0.5% stamp duty on the total cost of the shares and the brokers fee on top of that, when selling you don't pay stamp duty only the brokers fee.
To recommend a broker is difficult, as I'd need info as to whether your looking to trade in the short term or long term. How many companies stocks you'll be buying and how quickly you want to sell them, etc. Personally, I don't mind paying a little extra on the brokers fee. A lot of the cheap brokers only buy/sell at certain times of the day or week, which is no good if you want the price listed at the time.