Take the DLR from London City Airport to Bank. Follow the signs for the Northern Line and take a southbound train for one stop to London Bridge.
For Victoria, again take the DLR to Bank but then use the escalator connection to Monument, so that you can pick up the Circle or District Line from there.
If there's no train for Bank shown at London City Airport, you can take a Stratford International train and change at Canning Town.
http://content.tfl.gov.uk/dlr-route-map.pdf