Hi gilmoregirls
I moved across the river from North London 10 years ago and don't regret it one bit, it's a great place to live - loads going on, brilliant transport links despite not being on the tube, though of course all the places you've picked out are on the Overground system which is brilliant for getting to all sorts of places across London.
Dulwich village is lovely as you probably already know but of course very very pricey, Forest Hill is close and is an areas that seems to be very up and coming at the moment. It's a 15 minute bus ride from the station down to Dulwich Park and it's a quick walk across the park from there (10 mins?).
The nearest station to Dulwich Village is North Dulwich but there is no direct route from the areas you mentioned (correct me if I'm wrong anyone!), the closest direct link would be Crystal Palace station. Is that an area you would consider?
The P4 bus will drop you off slap bang in the middle of Dulwich Village. It's starts off in Lewisham and runs through Ladywell, Brockley, Honor Oak, bottom end of Forest Hill (Horniman Museum) and then the Village.
Another alternative would be to pick the P4 up at Dulwich College and it's about a 5 min ride from there or you could just walk it on a nice day. Dulwich College is a 10(ish) minute walk from West Dulwich station and trains run direct from Penge East (5 minute journey) or Kent House (7 mins).
I have a friend who lives in Penge and she has a lovely house (was a 2 bed but they've now had a full loft conversion to make it a 4 bed) with a nice garden. The street is off the high street and is really quiet and neighbourly. Let's be honest, the high street is a bit on the grotty side but really who cares and also who knows what's going to happen to this area. Most of the areas in SE London with good transport links are on the up rather than on the down.
Have you considered Peckham? Again, the high street is a bit of a dog but once you step back there's lots of lovely quiet leafy roads and great transport links at Peckham Rye station. Or Nunhead? lots of "bargains" to be had that way, much cheaper than a lot of areas in SE London and you're more likely to get a garden at a price you can afford.
are you living in SE London at the moment?
I think it's just a matter of getting the ball rolling and seeing as many places as possible. Most estate agents will cover a reasonably wide range of area so pick a good one, give them your criteria.
I'm in East Dulwich so don't have a huge amount of knowledge about the areas you've mentioned other than the village and Forest Hill. Penge/Sydenham/Brockley are in the Lewisham Borough so perhaps try posting on the Lewisham talkboard as well if you haven't already.
If you've got any more questions don't hesitate to ask!
Good Luck!