For public transport your best bet is to buy a so-called Travel Card for the amount of days you are here. You need one for Zone 1 and 2 (as London is divided in 6 zones and you don’t need to go to the outskirts)
This will give you access to and the tube, and the bus and most trains.
Not sure what it costs, but this is the best option.
Also look at http://www.tfl.gov.uk this is the website for public transport in London, very handy.
See you all over the next few days!
Cheers