Oyster card or travel card ?

I've never heard of the "Newton" area. Some estate agents are trying to get that part of town known as "midtown", but it isn't working.

The Oyster card is an electronic card that it is possible to charge with either pay-as-you-go money for travel, or with a Travelcard which gives you unlimited travel within certain zones. It is also possible to do both on the same card if you only want to make one or two journeys further out.

The Oyster system is meant to have a cap so that you won't be charged more than the price of a daily Travelcard, but this is not 100% reliable. If you are staying for five days it will probably be worth getting a seven day Travelcard which will be cheaper than five days at the limit. Another advantage of getting a seven-day Travelcard is that if you use PAYG or one day Travelcards you get charged extra for travelling before 0930 on Mon-Fri, but not with a longer period Travelcard.

You don't say if you intend to take journeys out of town or what transport means you are using to get to and from London. The only London airports that are within the Oyster/Travelcard area are Heathrow (Zone 6, covers the Piccadilly Line underground only and NOT Express or Connect rail services to Paddington station) and City Airport (Zone 3). If you are entering or leaving by Eurostar, St Pancras is within Zone 1. Most London tourist attractions are within Zone 1, the major exceptions including Greenwich, Kew Gardens, and the air museums at Hendon.

I am still confused but as Emma explained it looks complicated than what it actually is.
We have scheduled Justairports for the pick up from Heathrow to our apartment. We still have not decided how we are going to get to Gatwick airport for our flight to next city.
Among our itinerary is to visit Greenwich , Borough market, visit Bath (with London walks )and London tower. Thanks

The travelcard is also eligible for 2 for 1 attraction offers if you get the right one at the right booth. Read about it here:

http://www.tripadvisor.com/Travel-g186338-c195600/London:United-Kingdom:2.4.1.Travelcards.Step.By.Step.html