You can buy shares online. In fact, I think most people do now. There are many sites that you can use; Comdirect, Halifax, Barclays, T D Waterhouse, and many others. Their charging structures are different and it depends on how you are going to trade (frequency, size, etc) as to which broker would be the most economical.
Whether you make a profit depends on the shares you buy and future price movements which largely depends on future events. Because future events are random, no one knows whether prices will go up or down in the short term. But over a long term (5-10 years), shares will probably, but not necessarily, go up. Taking dealing costs into account, you have to be very lucky to make a profit in the short to medium term.
If you want to seriously learn a bit about investing before actually spending any money, I suggest you go onto www.fool.co.uk which is a site specialising in financial matters. It is free and you will get lots of advice if you ask.
If you are interested in trying to make a profit by picking shares that rise in value, look at the "value investing" forums. If you are interesting in making a profit by receiving income from share dividends, look at the "HYP investing forums".
Whichever route you take, only invest money you can afford to lose and remember that nobody (not even so called experts, journalists, etc.) can know the future movements of prices. It's all a gamble unless you have inside knowledge (which is now illegal)
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