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.