We (A family of three) will be having two days in London the end of December. We are planning to stay the city of London surrounding area. I guess we will use underground/buses or train for sightseeing. Is there any multi-days card to cover all public transportations?
Yes. It's called the Oyster Card. I used it in 2010, saved it and used it again in 2017. Here is background: https://tfl.gov.uk/travel-information/visiting-london/visitor-oyster-card