Well, a few words about London's public transport system - it's big, busy, blimmin' expensive to use, sweaty in the summer (NO A/C), varies from ramschackle to super efficient, but generally gets you around reasonably painlessly (except maybe in rush hour !!). See
www.tfl.gov.uk for inforamtion on tubes/buses/trams/trains/boats/taxis etc. Basically, the whole system is integrated in that you can buy one ticket for a day/week/month/or longer that will cover all forms of public transport. The city is divided into 6 concentric rings (Zones) - so you buy a ticket for how ever many zones you will cross on one journey), or else you just pay as you go, i.e. from station to station.
My advice would be to plan your location carefully to avoid long/difficult commutes, with the fewest changes possible - I speak from experience of living in Muswell Hill (North) and working in Kensington (West) - a 70 minute long epic walk/bus/tube/walk journey - I changed jobs! However, rents are highest around train/tube stations.
New Cross is in ZONE 2 on the East London tube line (funny little line that doesn't really go anywhere too useful - see Tube map!) and the North Kent overground line (run by South Eastern commuter trains, see
www.setrains.co.uk - very high frequency trains to London Bridge and Charing Cross in central London and out to the suburbs and onto North Kent). New Cross is also a bus hub, with buses coming in from all over SE London.
Places to look for a flat in SE London ... try the following ...
Greenwich - green, historic, very expensive, touristy in centre (but still pleasant)
Blackheath - ditto above, nice village feel, but still very much London
Rotherhithe/Canada Water/Surrey Quays/Docklands/Isle of Dogs - modern, watefront area, spacious - rents can be cheaper away from the main stations, but you might find a good bargain if you travel by bus
East Dulwich - Victorian area, mixed, good shops/cafes/bars - some bargains to be found?
Bermondsey - closer into centre of London, high density area, up and coming?
Woolwich - cheaper area, further out - some areas nice, others not so nice