Public transport in the UK is run by a variety of commercial companies and there are generally different fare structures and rules in place across different modes and networks. On the whole, though, children under the age of five years can travel for free, without any ticket or form of identity. There is usually a limit to how many children can travel with a single paying adult, that limit is four. In London, children aged 5 - 10 can continue to travel free on services provided by Transport for London, (buses and trams) but they may need an Oyster Zip card to prove that they are under 10.
Given that public transport is easy and cheap for little ones, why not consider taking them on an outing on a bus or a train? ...or why not see how many types of public transport vehicle you can ride in a day? Can you go on a regular bus, a 'double decker' and a train? If you live in the right area you might be able to add on an underground train or a tram even?
In London there are other fun and slightly different systems including the Docklands Light Railway, river buses and the brand new cable car cutting across the river between North Greenwich (O2 Arena) and Royal Victoria (near Excel). The river bus provides a wonderful way to travel from Docklands into the centre of London, with services as far out as Hampton, Kingston, Putney and Richmond. The river bus is by far the cheapest river ride available on the Thames.
Pack a picnic and plan a round trip using public transport. The children will love it and you will explore and see new places and experiences along the way.