You usually go to your GP who will literally just book you in as being pregnant, they ask you some questions about your health and then they liase with the local midwife for you. The doctors stuff is really simple and if you are 16 they have to keep your appointment competely private. Once they have registered you the midwife will contact you, usually by post so if you live at home this could be awkward until you have told your parents. I think the best thing to do would be to get to the doctor as soon as you can but explain to them that your family don't know yet, they may be able to get the midwife to contact you on your mobile or something if they know your situation.
I have to say though and I know its far easier said than done, but you really need to tell your mom! I'm 24 and I have never been that close with my mom but I still felt totally lost being pregnant until she knew, even though I am engaged to the father of my baby!! No body can be quite as supportive as your mom, even if she does take some time to calm down first!