I have got a quick informatio for people travelling to and from london they can book their airport transfer with HUMMINGBIRD CARS http://www.hummingbirdcars.com
They also provide 5 % discount on all return journeys
I'm pleased that you are due to visit London for the first time. December is a nice time to visit with all the Christmas lights and other things. There are National Express coaches which serve Stansted Airport, and they are probably the cheapest way of getting into the centre. For your return journey, you will probably find the train better as you have to be there early. You can take the Stansted Express from Liverpool Street Station.
I'm not sure about hotels as they all seem to be quite expensive. A friend of mine stayed at a hostel and it was really lovely and quite cheap too. Any of the areas which you mentioned are conveniently placed for sightseeing and you will be able to use buses or the Underground. An Oyster Card costs £3 to buy and then you can add some money to it to cover the cost of bus and Underground journeys. It works out much cheaper to use that. There are also Travelcards available. I would ask at an Underground station when you arrive in London as they will tell you which is the best option.