I wasn't brought up in Bromley but have lived in the borough for 5 years now in 4 different locations and probably know the area more than someone who's lived her all their lives!
I lived in Sundridge Park first which, for primary schools is okay, Scotts Park is a really nice school but then there's not many local shops though Bromley is practically a stones throw away.
I then moved to Bromley South which was more useful for transport links, local shops but I started a family here so then moved again so, personally, wouldn't recommend it but that's just me.
Then I was in Petts Wood, amazing area for me. Lovely, friendly neighbours, has a little town centre that does have almost everything you need. It is smack bang in the middle of a good catchment area for schools. When I was there my DD1 got in to Darrick Wood Nursery but just down the road you have Crofton too. Petts Wood is under Orpington but doesn't feel like you're in Orpington.
I'm now living in Bickley which is perfect for schools in my opinion. Only downside I'd say is comparing it to Petts Wood, local shops aren't as good.
All in all, Bromley is a lovely area. Well, coming from Camberwell anywhere could be a nice area lol. Area's I'd say to avoid are Downham, Grove Park, St Pauls Cray, St Mary Cray, Orpington Town Centre, Elmers End and Penge. But other areas like Beckenham, Hayes, Crofton, Chelsfield, Locksbottom, Farnborough, Elmstead Woods, Chislehurst, West Wickham and Shortlands are all lovely too.
Some people may disagree with my opinion on those areas so I'm sorry if I may offend anyone :)