Great advice above.
Public transportation in London - the Underground (Tube), buses, and some local rail - are operated by Transport for London. Visit their website and play with the journey planner once you have a trip in mind (from your B&B to the Tower of London, or from Victoria undeground station to the British Museum, etc.). As noted, you'll each buy an Oyster card. No need to buy the "Visitor Oyster" in advance, just buy 'em when you get there. The website is: https://tfl.gov.uk/
You must have some idea where to start because you've bought the plane tickets and booked the hotel. Take a deep breath, or three, and get your hands on a few guidebooks. Rick Steves, of course (hey, this is his website), and others such as Lonely Planet. Make a list of the things that each member of your family is interested in seeing/doing in London. Then start reading the guidebooks and refining the list.
These days, Google maps and the like are great resources to find the sights you want to see, in relation to each other and your hotel.
I've found - as have others - that a good way to plan your itinerary in London is to plan your daily schedule with geography in mind, visiting venues in the same area on the same day to avoid the hassle of long tube/bus trips in the middle of your sightseeing. (Don't ask me how I learned this the hard way because I shoulda known better. Okay, I'll fess up. I had to bug out early from a guided walk because I realized I had to get from down the river near Greenwich way back up to Westminster Palace to make it to a prebooked Parliament tour. Not a disaster but not fun either.)
Also, I have found - and others have too - that a good policy is to schedule no more than two "major" or "must see" things/places in a day. Those are the things that you absolutely will regret not seeing/doing, or venues that will take a bit of time to get through. Then depending on the time available you can fit in other things as time, interests, weather, and life's general unpredictability allow.
We have no idea what your family's "must sees" would be. You could be serious outdoor hikers and not interested in art museums, or a bunch of serious musicians, or railroad geeks, or diehard Beatles fans, or . . . anything else! So make some lists of what you'd like, and have fun planning. I love London and I'm sure you'll have a good time!