I've cycled on big city roads for years...almost 20 years, actually. Am a very considerate and law-abiding cyclist and have only had one minor accident in that time - stationary car on quiet road, driver opening door without checking.
I've mostly cycled all over London, by the way. And women cyclists get treated better by other road traffic than men cyclists.
If you want to cycle I think it's a great idea, but it'll take you some time to get used to it so be patient with yourself. I was very afraid to start with, for quite a long time, but now I'm assertive on the road - an essential skill. You'll learn how important it is to place yourself ahead of the cars at junctions, how to hang back behind large vehicles for safety (especially on bends) - IME these are two of the most important lessons to learn, plus make sure you are visible to other drivers - catch the drivers' eye before passing in front of them, for example.
Also remember that even though a cyclist is not in a motorised vehicle, you are still traffic; i.e. you have as much right to be on the road as any vehicle. A lot of drivers are too stuck in the motor world to realise that - it makes me snort to see some of the anti-cycling comments on this thread, because, unlike a bad cyclist, a bad driver has a ton of metal behind them. In fact, in this day and age I would go so far as to say that any emission-free vehicle has MORE right to be on the road than one spitting out fumes.
Anyway, definitely practise on quiet roads and cycle paths, get reflective gear, good lights, a helmet (pay around £50 and make sure you wear it horizontally on your head, i.e. not tipping upwards above your forehead and down towards the nape of your neck. Also make sure the straps are tightened so that when you shake your head the helmet stays more or less still), and a good lock (pay about 10% of the value of your bike; make sure you lock the wheel to the frame, and don't get a Kryptonite lock as they can be easily compromised by a bike thief. Abus do good locks).
If you contact a local cycling group (I think Sustrans might be able to help you find one) you might find cycle tours in your area - basically historical tours where a group of cyclists bike around the streets with a historical guide. It's great for building a sense of confidence as a cyclist.
Best of luck, PSM!