Oyster card or 7-Day Travel card?

Oyster is an electronic wallet rather than a discount travel plan. Deposit some money in it and then travel. It opens the gate for entry, and on exit it deducts the appropriate fare (the Underground has zone prices.) There is a top limit per day after which trips are free. It's the easiest way to use the Tube where impatient commuters don't want to stand behind you as you fumble with a pass.

For buses you tap in but not out.

Transport for London:
https://tfl.gov.uk/fares/how-to-pay-and-where-to-buy-tickets-and-oyster/pay-as-you-go/oyster-pay-as-you-go?intcmp=55536

Use this site to plan your journeys too.

The issue with a Travelcard is you have to buy the one that covers all the zones of the Tube you will be crossing into. The more zones the card allows, the more expensive it gets. If you cross into a zone the purchased Travelcard does not cover, you cannot exit the Tube without paying an extra fee. Depending on where you are, there may not be anyone there to collect the fee so you cannot exit!

With the Oyster, it takes care of all the calculating. If you stay in zones 1 & 2, where most touristy sites in London are located, you pay no more than you would for a Zone 1 & 2 Travelcard. If you stray into another zone, the Oyster covers it and you pay for the extra zone charges only on the days you go into them. Just make sure you add more money to the Oyster as it draws down so you don't run out. The good thing is with the Oyster, it can be topped off with any amount you like at most any Tube ticket machine, no staff assistance required.

To confuse things even more, you can buy Travelcards and have them added electronically to your Oyster. So when the Travelcard daily max is less than the actual trips you took, that is the max that is applied. Once again, you probably want to keep this to the minimum zones you will cover on most days to keep the cost to the minimum. No point in paying for a 6 zone Travelcard if you only go that far one of the 7 days.

Are you traveling solo?

When I am in London for a week or more, I usually load a zone 1-2 travelcard on my Oyster with some extra pay-as-you-go money to cover trips beyond zone 2 or extra days. The daily cap for zone 1-2 is 7 GBP whereas the weekly card is 35.10 GBP. Since I usually hit the cap everyday, the weekly travelcard saves me some money over using just pay-as-you-go.

It only takes 3 trips in zone 1 to hit the daily cap.

Note that Windsor is outside the Oyster/Travelcard area, even though it isn't as far as Leicester. No point in buying tickets in advance though.

If you cross into a zone the purchased Travelcard does not cover, you cannot exit the Tube without paying an extra fee. Depending on where you are, there may not be anyone there to collect the fee so you cannot exit!

You are supposed to buy a ticket from the boundary of the zone your travelcard covers to the destination before you leave.

Note that you can do this to Windsor and for that matter Leicester as well, although the reduction in fare with a Z1-2 TC will be small.

Thank you so much for all the info and tips!! I'm still a bit confused but will hopefully figure it out by the time I get there. :) We're also happy to pay for convenience so we don't have to worry about calculating things as we go. It's just my husband and I travelling and we just like to keep things simple.

Marco's point is important. It's not true that you have to buy a Travelcard to cover all the zones you will travel in. It is true that if you are traveling outside the zone limits of your Travelcard, you have to pay the extra fare in advance. If you have a paper Travelcard, you have to get this as a ticket; if you have your Travelcard on an Oyster, you can add the money to the Oyster.

Once upon a time, if you went beyond the limits of your ticket or Travelcard, you could pay the "excess fare" upon exiting with no penalty (the way you still can in Washington DC and Tokyo). Now, in London, you are considered to be "traveling without a valid ticket" if you do this, and you can get fined.

For instance, I bought a 7 day, two zone Travelcard on an Oyster at Heathrow, but wanted to take the tube in from Heathrow (so, zone 6 to zone 1). I was traveling off peak, so the extra fare for this was £1.50 (I forget the peak supplement amount). I merely added £1.50 to my Oyster. I then tapped in and tapped out as normal, with no problems. In my case, that was my only trip outside zones 1 and 2, so it was easiest and cheapest to do this.

So, if you do get a Travelcard, you can just get a two zone one, and add some cash to it to cover the extra charges for your trips to Hampton Court Palace and the like. Or, you can just get an Oyster and put on money, then top up as needed.

Since there are two of you, there are some potential savings (2for1 offers) by using a paper travelcard issued at a rail station.

https://www.daysoutguide.co.uk/2for1-london

You would have to review the offers to see if any of the sites you plan to visit apply for your dates of travel. The paper travelcard needs ro be bought at one of the major train stations (Victoria is closest to your location) and require a photo. If you choose this route, you could get an Oyster to cover your extra day.