I live on the other side of the borough from Catford. It's a mixed area but the flats by the shopping centre are scheduled to be demolished and that will make a big difference. There is a huge new estate being built on the site of the old greyhound track very near the stations which will also change the demographics. I like living in SE London. I've been here for nearly 30 years and it's changed a lot in that time.
I'm a long way past the baby/toddler stage so can't tell you what's available on that front but I'd be amazed if there isn't a fair amount. SE London seems to be going through a huge baby boom!
There are lots of green spaces nearby - check out Beckenham Place Park, Ladywell Fields, Blythe Hill Fields, Hilly Fields, Mountsfield Park, Manor House Gardens, Manor Park, Telegraph Hill.
Points to consider:
London Borough of Lewisham, solid Labour (almost no councillors who aren't Labour) - fairly high council tax, services more of a mixed bag. Refuse collection is good, libraries and pools not bad.
Shopping in Catford - not brilliant, but you would be very close to Bromley and Lewisham for better shopping (Bromley is a lot more upmarket than Lewisham but Lewisham has all the basics including a good street market).
In my experience Lewisham Hospital is generally good. It's a teaching hospital with (currently) A&E, full maternity services and a SCBU.
Transport links are a big bonus. Two railway stations in Catford and others nearby. Lewisham has the DLR as well as National Rail. The Overground runs south from New Cross Gate to Croydon and the stations are not far from Catford. Lots of buses. (At some point the Bakerloo line is likely to be extended to Lewisham and possibly Catford, but I wouldn't hold your breath over that happening within the next 20 years.)
Lewisham schools have improved a lot but the secondary schools are a more mixed bunch than the primary schools. Within striking distance from Catford you would have Sedgehill - mixed comp, on the up; Trinity - C of E small mixed comp, on the up; Coniston, mixed comp, on the up; Prendergast Ladywell Fields, mixed comp, on the up (possibly more of a bumpy ascent than some of the others); Forest Hill Boys, comp, consistently well regarded for a long time; Sydenham Girls, comp, ditto; Haberdashers' Aske's Knights, mixed comp, on the up as far as I know.
Further out: Bromley schools are generally good, including St Olave's and Newstead Woods, which are super-selective grammar schools; Bexley and Sutton grammar schools; Dartford grammar schools.
Private schools: St Dunstan's, consistently well spoken of; Blackheath Girls'; Eltham College; the innumerable schools in Dulwich.
Hope this helps!