The first step is to choose and register with a broker. If buying UK stocks, you should choose one that is a member of the London Stock Exchange. Some brokers offer online trading, others charge more and give advice.
http://www.londonstockexchange.com/en-gb/pricesnews/education/resources/broker/
It's worth having a browse around other parts of that site.
Once you've registered, you just follow the broker's system. If it's an online service, you just log in to the site, get a quote and click in the right place to buy or sell.
A little advice for a new investor:
• Before using real money, try just pretending. Pick your share and follow the price to see if, had you bought, you would have made a profit or loss.
• Take all recommendations with a pinch of salt and make up your own mind. Ignore any hot tips for companies that you've never heard of before and, particularly, avoid anything that is being recommended solely on the basis that the price is a few pence rather than a few pounds "so it must be better value". Recommendations from a newsagent's shelf are generally better and cost a lot less than subscription "tip sheets".
• If you want to let someone else make the decisions, choose from unit trusts and investment trusts. Many other forms of equity investment have higher charges and complicated terms.
• If you get an unexpected phone call from someone with a "not to be missed" investment offer, either hang up or ignore their questions and ask for details of their name, their company's name, address and so on and you can be sure that they will then hang up.
• Don't be rushed into anything. There are opportunities today but no more and no less than the opportunities that there will be tomorrow.