Thanks for the additional information. Other posters will add their thoughts (I expect!) but I'll speak from my experience.
My first trip to London was last December with 3.5 days for sightseeing. Like you, I prefer not to have "time on my hands". I would rather have slightly overscheduled and then cut something out than to take up precious time trying to figure out what else I could see or do.
With only 3.5 days, stay in London--save day trips for another visit. Plan on using public transport--tube and buses. The system works very well. I found the Citymapper app to be very helpful in planning journeys.
Get a good guide book, Rick Steves London or Lonely Planet London are two I like. Check opening and closing times of sights you're interested in. For example, on Sundays, both Westminster Abbey and St Paul's Cathedral are open for worship services only (visitors welcome) but closed for tourist visits. Certain evenings, especially Fridays, some museums are open late .
Get a good map. I like Benson's Mini Map--it costs less than $3 and they ship for free: http://www.bensonsmaps.co.uk/our-publications/london-mini-map/. It shows points of interest, bus routes, theatres, etc. So much better than Google maps for planning your itinerary. See what sights are near each other to plan your daily itinerary.
For a first visit, I'd recommend considering these:
Tower of London 1/2 day at minimum, get there a little before opening, once inside go straight to the Crown Jewels while the line is short or nonexistent, then circle back and join a free Yeoman Warder tour.
Westminster Abbey 2 hours minimum or go there for Evensong prayer in the afternoon if you are not interested in touring the whole thing
Boat ride on the Thames even for just 30 min, because it's wonderful to see London from the river. There are (longer) tourist cruises with commentary or you could just take a commuter boat (Thames Clipper) with no commentary. Depending on where you are staying, you could take the Thames Clipper (http://www.thamesclippers.com/) from Westminster or London Eye Pier to the Tower of London or vice versa.
Museums pop inside one of the free museums for an hour or two, but plan ahead what you want to see to avoid being overwhelmed. British Museum if you want to see important artifacts from around the world, National Gallery for art masterpieces, Victoria and Albert museum for the decorative arts.
Parks London has wonderful green space! Hyde Park, Green Park, Kensington Gardens, Regents Park, Hampstead Heath, and more. See what's near where you'll be staying.
Walking tours . Walking is the best way to see London. You could follow one of Rick's self-guided walking tours, or go with a company like London Walks (no booking ahead, just show up at the appointed place and time and pay in cash. http://www.walks.com ) . I've taken several LW tours and have not been disappointed. Edit: in light of the fact that your 12-year old doesn't like walking, either skip this entirely or let the 12 year old choose a walking tour that appeals him/her to make it more interesting (Harry Potter Locations, Spies and Spycatchers, etc).
Food you'll be "spoilt for choice" as they say, with all price ranges, all cuisines to choose from. I'll just mention Borough Market for a variety of food stalls (Wednesday-Saturday is full market, crazy crowded on Fri/Sat), Duke of York Square food market near Sloane Square on Saturdays, and Camden Market/Camden Locks Market, if one of these is near where you'll be staying or sightseeing.
I'd skip the Changing of the Guard at Buckingham Palace (it's a time sink) but if you like this kind of thing a good alternative is the Changing of the Horse Guard at Horseguards Parade).
I could go on and on but I'll stop now. I hope you have an enjoyable visit!