Are you staying in centralish London. In which case I wouldn't worry about ordering a cab.
Cheapest way to get around is to buy a travelcard ( there is off peak and peak time variations depending on times you wish to travel) .
Don't pay for single journeys on the tube as they are a ridiculous price £4
Check what lines are open that weekend though if you are needing to go to certain places - weekend engineering works are a constant source of pleasure
http://www.tfl.gov.uk/This should give you all your info