I've been using Hargreaves Lansdown for 16 years and can highly recommend them. If you're not sure what to invest in, HL offers ready-made portfolios tailored to your risk profile (you take a profiling quiz to establish your attitude to risk).
HL are not the cheapest if you have a lot of money to invest and will hold a large portfolio, but they're very competitive for the average, smaller investor. By small, I mean if your investment is less than about £60k.
As others have said, you could opt to hold tracker funds, which have very low charges, typically 0.03%. The alternatives are managed funds, which have ongoing management fees ranging typically from 0.5% up to 1.75%, or buying shares direct, for which you will be charged a transaction fee and stamp duty for each trade.
Holding individual shares is quite high risk, so never put all your money into just one company, choose several.
Managed funds do the hard work for you and invest in a range of equities and products within the objectives of the fund. A good fund manager should be able to beat the underlying index and give a better return than a tracker, but of course even the best managers can from time to time make bad decisions, so there are no guarantees.
The thing to always bear in mind is that investments can go down as well as up, so you need a long-term investment horizon (your suggested 10 years is good) in order to benefit from the (one hopes!) general upward trend over time, whilst riding out the dips and troughs.