The Underground and all buses don't run on Christmas Day, taxis charge a large premium (probably double or triple fare). Normally the best way to get around is by Underground or Bus, if you buy an Oyster card and load it with prepay credit there's a daily limit so you'll never be charged more than that (just make sure you touch in and out) especially if you just stay in central London. You can pick up an Oyster Card at any underground station, there's a £3 deposit but this is returnable once you return your card. There's lots of Tourist information offices around London.
The nearest station to Big Ben is Westminster and this is also the nearest station to Westminster Abbey, Downing Street and is within walking distance of Buckingham Palace.
There are a number of tours by bus:
Big Bus tours -
http://www.bigbustours.com/eng/london/de…
London Duck Tours - both on land and the River -
http://www.londonducktours.co.uk/
The Original London Sightseeing Tour -
http://www.theoriginaltour.com/
Please note that these tours will cost extra on top of other travel and you will not get a discount with an Oyster Card.