There is no real centre for the area, however each neighbourhood tends to define its centre by the local station / parade of shops. Having said that, I don't think that there are any areas you should avoid around here and there is little difference between being east or west of the railway. This part of London is well served by open spaces with Sydenham Wells Park, Mayow Park, the Horniman gardens, Sydenham Hill Wood, Dulwich Park, One Tree Hill, Honor Oak Sports Ground, Blythe Hill, Peckham Rye and other smaller spaces.

The main advantage of commuting to Forest Hill used to be (a) that you could get on the train in the morning (although sardines in a tin had more space) and (b) trains used to run fast to Forest Hill. Neither of these apply anymore since the arrival of London Overground in May. However, Forest Hill arguably has better bus service.

One aspect you might like to consider (if a night owl / late night reveller) is the proximity of a night bus.