Basically you just touch in and out using the bank card (or device like a smart watch or phone). You must use the same card/device each time. The system analyses all your touches at the end of each day and makes a charge to your bank for the cheapest fare for the days travel (often will be the cap).
You don’t have to do anything else, it just works. You can create an account on the TfL website and link the card to it so you can check your journey history if you want, but it’s not essential.
The 10-year-old travels free on all buses, tubes and dlr. If you’re sticking to zones 1 and 2 then you shouldn’t need to use any National Rail so they shouldn’t need a ticket at all. The map at https://content.tfl.gov.uk/free-travel-areas-without-oyster-adult-accompanying-map.pdf shows where they can travel for free on National Rail without a zip card. Any of the green lines are fine.