I went on the pill for the same reasons and Im also quite shy and easily embarrassed but its not as bad as you think.
Just book an appointment with your GP, maybe ask to see a female GP if you prefer, and just tell them that you would to go on the pill. You can go on your own or bring someone along if you like.
They'll ask you a few questions like have you been on it before, are you in a relationship, explain how and what you do. My GP printed off some info for me too so you can ask them to do that for you too. They may also take your blood pressure too.
They'll probably put you on Microgyen first because thats the cheapest one available on the NHS but if you don't like being on that one, and find yourself having side effects, then just go back and ask to be put on another one. Thats what happened to me and this new one is much better. Everyone reacts differently to different ones.
All in all, it probably took 10/15 mins at most (appointment wise) and you just go back every 6 months for a check up.