The World is Your Oyster
A prospective visitor asks: "What do you use for public transit--oyster card? bus/train passes? Are you zone 2 or 3? we are thinking 7 day pass"
The Oyster card is a marvel of technology.
We use the Oyster card with cash (not Travelcard) and automatically top-it-up with cash when it falls below £5.00 (it is automagically debited from our bank account).
We are Zone 2 (Putney Bridge is the nearest tube stop, Putney railway station is the nearest train stop). The buses are flat fare throughout the city. The national rail system is inconsistently on Oyster, but all of the buses and tubes are, and it works well, and is guaranteed to be as cheap as the one-day Travelcard alternative (it has automatic price capping so if your total one day travel exceeds what a day travelcard would be, you only pay that) assuming you don't use national rail. Also there is bus price-capping, so if you spend more than £3 on bus travel, the rest of your bus travel is free that day. The buses work very well now in the central city with the congestion charge taking out most of the private vehicles. They are still slowed by excess traffic in Kensington and Chelsea (and other areas), but the congestion charge is expanding to Kensington next year. The bus frequency is high and the signage excellent (especially compared to the Twin Cities).
Apparently the three day Travelcard might be slightly cheaper depending on your usage if you stay in zones 1-2. If you are planning on going beyond that, the three day travel card price = 3 * the one day travel card price. (Heathrow is in Zone 6, Gatwick is not Oyster-compliant, you still need special tickets for the Gatwick Express). The Travelcard is also useful if you are using National rail within the city (we seldom do), as National rail is not fully Oysterized.
The seven-day travel card for Zones 1-6 is £41, for Zones 1-2 it is £22.20. In principle this is a discount over the maximum daily (as it should be). However, on my daily travels throughout the city, I wind up spending a little over £20 per week (but not every day involves travel by tube, and some of the travel is off-peak). As tourists you may spend more (say up to £6.00 per day for 2 tube and 2 bus segments per day, in which case a Travelcard is much cheaper) I stay mostly within zones 1-2 (except for Heathrow). (Note travel originating and destined for Zone 2 may be cheaper than Zone 1 to Zone 2, even if you pass through zone 1 to get there, depending on the presence of alternative routes). You can always get the seven-day travel card on Oyster and add cash for travel outside Zones 1-6, it is supposed to be smart enough to charge the right amount (I have not tested this particular claim).
It does cost £3 to get the card itself, but it pays for itself in 4 bus segments or 2 underground segments compared to cash tickets typically. You can put the electronic Travelcard on the Oyster card, or just top the Oyster card up with cash.
Oyster cards are sold at most (all?) tube stations and many convenience stores.
Oyster also has some associated coupons (which we have yet to exploit), and you can get refunded the cash balance when you leave if you want.
My suggestion is get an Oyster card and put a seven-day zone 1-2 travel card on it along with some cash if you plan to take the Piccadilly Line from/to Heathrow. If you wind up spending more, you can top up while you are here. It will still be good when you leave (my guess is the technology is stable for about a decade ... this is London so there will probably be Oyster readers a century from now, however your particular card may deteriorate somehow), so you can bring it when you return to London
While the technology is clear, the fare structure is still quite complicated, as befits a system this large and convoluted.
The details are here