OK, thereinmadnesslies (are we related? 
), the company I use is the largest in the field, Hargreaves Lansdown. Another highly reputable company in the sector is Chelsea Financial Services. If you Google "Fund Supermarket" you will find others, although not all of them are fully independent or cover a wide range of funds, so choose carefully. I particularly like the manageability of the HL site, and DH moved his savings from CFS to HL for the same reason.
It is DIY investment unless you have savings of over a certain amount (I think £100k is when some level of personal advice can kick in if you need it), but the website contains huge amounts of information to get you started. They have suggested portfolios for low, medium and high risk which you can use to get started with if you want www.hl.co.uk/investment-services/isa/ready-made-isa , and also their "Wealth 150", which are their top recommended funds.
I tend to look at those first, but if I'm trying to balance my portfolio by sector or region, I might look at other funds and make my own decisions based on past history and the wealth of research provided on each fund, plus ideas from reading the financial sections of the weekend newspapers.
You can also set up "watchlists", which were previously known as a Virtual Portfolio, and add mythical sums to it to see how they perform over time, before actually investing.
I've only had my fingers burned in a very small way twice. One was a very high risk investment in the 3rd world where I put in £2,000, just as a toe in the water, and lost about £250. The other was some shares I received from a demutualisation donkeys' years ago, so in one sense I didn't actually "lose" any money, but I could have timed the sale of them much better.
That's why I only invest in funds not shares: funds are essentially just bundles of shares and other investments, where the fund manager is doing the research and constantly aiming to improve the return for his investors by buying and selling.
Your basic starting point is to open a Stocks & Shares ISA (assuming you haven't used your £20k allowance this year) and then "go shopping" with either real or virtual money. I tend to start with £2k/£3k maximum in a fund and add more later if it seems to be going well. If you've used your ISA allowance, you can still invest, but the returns will be taxable until you can move them into the shelter of an ISA.
It can be good fun, and is quite addictive at first, but as I said up thread, I don't review more than every 3 months because it's too easy to lose your nerve when markets get volatile, and continuous buying and selling can get expensive. It is definitely medium to long term investment.
Hope that helps!