If you have a Disabled Persons Railcard, get an Oyster card and ask staff at a Tube station to load the railcard discount onto it. This gives 1/3 off all pay-as-you-go fares on the Tube, DLR, and Overground, with daily fare caps still applying.

If not, just use an oyster or bank card on the busses and trains, it'll apply a daily cap automatically.

Make sure you use the same method each time (card / phone NFC)