I've lived in Forest Hill for 3 years now and I really like it! I'm expecting my first baby in April and I'm sure I'll find plenty of things to do with the baby. I've signed up for NCT classes which are in Catford because FH doesn't have a group, but Catford is really close.
The rail links are fine (and the tube extension has been approved if you can wait that long!) and there are lots of buses from central london which run all night, or you can pick it up from Oval/Vauxhall etc. Cabs home cost £20-£30 depending on where you go from.
The shops aren't up to much at the moment but Sainsburys have FINALLY got planning permission to rebuild their store which is starting now and should be done in a year (it's dire at the moment) but will have a deli, bakery etc. And there is a massive savacentre just down the road in sydenham anyway.
The Horniman Museum is fantastic and has loads of stuff for kids to do - workshops, summer schools, special exhibitions, the hand on base etc, plus it has a completely child-friendly cafe and beautiful gardens for picnics. There is a playground opposite, and Dulwich Park and crystal Palace Park are also close.
There has been a lot of work recently on several pubs which have been refurbished and we now have the Dartmouth Arms which is a sort of gastropub where a lot of people take kids for lunch at weekends, plus lots of other places to eat. And East Dulwich is only just up the road. And there's the swimming baths, library...sorry! beginning to sound liek an estate agent!
I like it anyway, and have never had any problems with crime in the area. But I'll probably move in a few years anyway to somewhere a bit more green and a bit more Northern!