You can download details of ticket prices for London buses and tube trains here:
http://www.tfl.gov.uk/tickets/faresandti…
If you go to page 27 of the guide you will see the cost of a one week travelcard. A London Pass with travel is £38 more than one without travel and for that you get the equivalent of a zone 1-6 travelcard for 6 days. Note that a regular 1 week zone 1-6 travelcard is £44.60 so this is hardly a great saving. In fact you don't need a zone 1-6 travelcard, most of the tourist attractions, with the exception of Heathrow airport and Wembley, are in zones 1 and 2, so a zone 1-2 travelcard (cost £24.20) will do the job. (You can save even more money if you do most travelling by bus rather than tube as a one week bus ticket which will give unlimited travel anywhere in London is only £13.)
Note also that Gatwick is well outside London and is not covered by the travelcard system.
Residents of the UK can load a one week travelcard onto their Oyster, but this is not a facility available to overseas visitors, as it requires the Oyster to be registered to a UK address. However, you can get an Oyster and use it unregistered for 'pay as you go'. The previous answerer who said it takes a week to get an Oyster is talking rubbish - it takes less than 5 minutes from any underground station or shops that are agents for Oyster.
So this is my suggestion:
When you arrive at Heathrow go to the Underground station and get an Oyster Card. Put on enough prepay to cover your journey to the hotel - a single journey on Oyster is £3.50 Monday to Friday daytime and £2 other times - plus the £3 (refundable) deposit. (Page 9 of the ticket guide.) Also get a one week travelcard for zones 1-2 (or a one week bus pass depending on how you intend to travel). This can be postdated, for example if you are not intending to do any further travelling they day you arrive in the UK you could get the travelcard to start the next day. If you do find yourself travelling outside central London, use the Oyster and buy extra prepay as you go along.
For your journey to Gatwick just buy a ticket at the railway station - London Bridge station will probably be most convenient for you.
Have a great trip and hope this helps.
