Echo, the answer to your questions is in the link you provided (though perhaps it requires someone familiar with London to recognise the information)
a short journey by London Overground or train, from Crystal Palace railway station (which is only 7 minutes walk from the house); or by direct buses from Crystal Palace bus station (5 minutes walk from the house).
So, you just buy an Oyster Card and fill it up with some cash. Think of it as an "access all areas" pass for all public transport in London.
Then, depending on your mood, you either get a train from Crystal Palace station, or a bus (slower, but you get to see more) to wherever you want to go in London.
This is a map of all London's rail lines - tube and national rail: http://content.tfl.gov.uk/london-rail-and-tube-services-map.pdf
You can see Crystal Palace in the middle of square E5. It is at the end of the orange line, which is an orbital line, and on the green-and-white line, which gets you to London Bridge in one direction, and Victoria in the other.