Getting around London....so confused

The "plan a journey" feature is pretty helpful:
http://www.tfl.gov.uk/
The journey planner combines modes (e.g. walk, bus, tube etc) to get form any point A to any point B at any day and time in the future. You can even enter "Luna Simone Hotel" in the "From" box and "Hampton Court" in the "To" box, day and time of departure and it will take care of the rest.

Check this post out by a local in London.

https://community.ricksteves.com/travel-forum/england/using-london-buses-advice-from-a-local-it-s-a-bit-of-a-long-one-sorry

It is not much different from when I was in London in '63 & '73.

Drop the poster a note if you have questions.

My 2-cents worth.

Buy an Oyster card and stop worrying. Buy it at Heathrow before you get on the tube, and put £20 or £30 on it. Or you can buy one before you arrive.
An Oyster Card is just a smart card on which you either put credit, or other tickets such as a 7-day travelcard. You don't need a photo or anything. Scan the card at the turnstile whenever you enter or exit the tube or other railway station, scan it when you get on a bus. The card will stop charging you when you reach the cost of a Daily Travelcard, and either way is about 50% of the cost of cash tickets - you can't loose.
When you scan the card it says how much is left, when it starts getting low add more cash. You can do this at ticket offices, ticket machines and many newsagents. All the places you list (Greenwich, Hampton Court. Tower of London and Eye) are within the validity area of an Oyster Card.

Your Grandaughter doesn't need a card if she is travelling with you. "Children aged between 5 and 10 can travel free on Tube, DLR and London Overground services when accompanied by an adult (up to four children per adult)". If she looks over 10, carry her passport as proof.
When you leave you can either cash in your card, or save it for your next trip.

Must agree with Chris and just get an Oyster card. Let me add that your eight-year-old granddaughter would probably really enjoy taking the Thames cruise to Greenwich.

Here is the site that lists the places you can get 2 for 1 adult entry. Buy two adult tickets together, then purchase the child ticket separately. You need to purchase 1-day or 7-day paper Travelcards from a National Rail (train) station because it's their promotion. For the 7 day cards, you will need a passport sized picture that you can print at home.

If you don't want to use the 2 for 1 offers, I would get each adult a 7-day Travelcard for zones 1-2 on Oyster. Add some PAYG money to cover travel to and from Heathrow, day 8, and out of zone travel to Hampton Court Palace if you plan to go there. 7 or 8 days of PAYG on Oyster is probably more expensive than the flat rate 31.40 GBP for a weekly Travelcard. Your granddaughter will travel for free, unless you take a National Rail service to Hampton Court Palace, then she will need her own ticket.

In case you didn't know.... There is a bus stop right in front of the Luna Simone.

Judy, everybody has a different vacation because everybody is different, and we all are interested in different things.

As you will have seen in your previous trips most big museums are free, and most attractions charge an admission charge. Hampton Court. Tower of London and Eye for sure charge admission.

What you need to do is look at the Days Out promotion for the places that you are sure you are going to visit, see if they are listed, and see what savings are available. I presume that it is 2 adults and one 8 year old? And that all 3 of you will want to visit all the places you are considering? (you don't say how many "we" is)

Then consider how many and what kind of rides you will using, and do the arithmetic.

Certainly it will be cheaper than many other options if you have a weekly Zones 1 and 2 Travelcard. That can either be a paper one from a National Rail station or on an Oyster card from a Transport for London outlet.

If it is a paper one you will need another method of paying for your travel outside zone 2. That could be Heathrow (Zone 6), Hampton Court (also Zone 6), etc. Greenwich is on the border of Zone 2 and Zone 3 and either is correct. To go beyond, and to arrive the first day, you can pay cash fares (very expensive) or a Pay As You Go Oyster Card for each adult person loaded with sufficient cash to pay for those journeys. Those prices are available on the TfL website. If you choose to use to use the Heathrow Connect or Heathrow Express you need different tickets for them, Oyster Cards and Travelcards are not valid on those.

If you get a paper weekly Travelcard from the railway so that you can save using the Days Out promotion you each need photos because the railway uses photo ID with season tickets which a one week Travelcard is. If you get a one week Travelcard on your Oyster Card you can also put the Pay As You Go money on there too - the system works out the cheapest way to charge - and you don't need a photocard, but you can't do the 2 for 1 at Days Out.

If you take the granddaughter on the buses or Tube she is free. If you take her on a train she is half price and needs a Child ticket.

You need to do the maths and look at the aggravation factors. Is the small savings overall worth the hoops you have jump through? Only you can decide.

Whatever you do I hope it is enjoyable for all the right reasons for all 3 of you.

By that time direct use of contactless cards would have just come in for payment - but I think there would need to be some practical experience with US cards, particularly Visa ones (MC is known not to work, and Amex should do) to recommend this approach. These will be capped to the price of a weekly travelcard on a Monday to Friday basis.

WOW! What wonderful and helpful replies. Thanks so much. I think we will just go with the Oyster card and make it simple. We'll debate about Hampton Court as there are so many sights to see. We plan to go to 2 matinees...Matilda and Lion King. Is it smart to reserve ahead for those or just get tickets when there? London has so many wonderful sights for kids. We'll ride the bus when possible and the bus by the hotel is a plus.

Hi

just as an aside, I believe that your grand-daughter would probably love to see the Crown Jewels in the Tower of London - an expensive venue to get into, but I took my 8 year old niece there a year or two ago and although she thought the Tower a bit (although her younger brother was enthralled!), her eyes popped out on stalks when she saw real crowns with huge diamond, orbs, sceptres etc etc....

Definitely book now for the theatre. Well-known musicals are often sold out on the day.