Full information on children's ticketing is on the
TfL website here - click the relevant age range tab at the top of the page.
Your 8 year old will travel with you for free on the Tube and buses (on suburban rail in most cases children aged 5 and older have to pay).
Your 12 year old will need to pay. To benefit from child fares they'll either need to apply for an 11-15 Oyster photocard (which costs £10) and then pick it up, with proof of age, from one of TfL's six
Travel Information Centres on arrival in London (as detailed under the "
Non UK customers" heading
here). The 11-15 Oyster photocard affords the holder free travel on London buses, and child-rate travel on the Tube and suburban rail services.
The other option is to buy them a child Day Travelcard - the off-peak version is priced at £3.20 (can only be used after 09:30am on weekdays, covers all of zones 1-6), the 'Anytime' (i.e. peak) version is £4.30 (and covers zones 1&2 - more expensive versions are available that cover more zones). These cannot be purchased from self-service ticket machines at Tube stations, one must go to the ticket counter - however you can buy them for several days in advance in one transaction. These Day Travelcards would afford them unlimited travel on the Tube, buses and suburban rail within zones 1-6 in London.
The third option is for them to travel as an adult and so pay the adult fares.