It isn’t simple when you are traveling with kids. They are entitled to discounts (the 16-year -old) or free travel (the 12-year-old) on London buses, trams, the Tube, and the Uber Clipper boats, but they each need a Zip Oyster Card as photo ID. You can read how to obtain those here:
For the parents, contactless payment with a “tap and go”:credit card is quick and easy, and most likely cheapest as well due to the “daily cap”.
If you want to travel as cheaply as possible around London, that would be by bus, where the adult fare is only £1.65. It is also a lot more fun thar nixing the noisy underground tube, especially if you head upstairs on a double-decker bus. It is slower than the tube, but for short journeys maybe not, especially if there are bus stops at or near your point of origin and your destination. Here is a schematic map of the main London bus routes and the major sights they serve:
As for getting from Heathrow to your hotel and back after your trip, the options are Tube ( including the Elizabeth Line), the Heathrow Express, or car service.
Our strong preference—-and I admit we are outliers here—- is for the fast and comfortable Heathrow Express. People say it is “too expensive” but if you buy your tix 90+days in advance they are only £5.50, cheaper than the Tube ( and kids 15 and under are free). The other objection is that it only goes to Paddington Station, which is true. When we are making a short stay in London, we stay near Paddington and walk to our hotel. For longer stays (more than 2 nights) we take a ride on a black cab ( from the taxi stand right outside the door to the street) and enjoy the sights as we drive through London. You, and especially your kids, might enjoy it too. It has cost between £15-20 when we have done this, depending on the location of your hotel.
Here is more information on the Heathrow Express:
Many people here will advise you to take the Tube, especially if your hotel is near the Piccadilly Line (or now the Elizabeth Line as well). I suspect many of those advisors are traveling solo or as a couple, which is easier than traveling with a family and luggage. My objection to the Tube, apart from the fact that it goes underground and is noisy, is that it is crowded—-too crowded with luggage. There is no place to put it other than to hold it by your feet. The advocates will point out that you board an empty car at Heathrow so will have no trouble finding seats. True, but the one time we did take the Tube ( because the HEx was not running), the car quickly filled up with people standing bumper to bumper in the aisles and blocking the doors so it was difficult to exit, especially with luggage. It was not a pleasant experience and we won’t do it again.