Hi! Harborne also has beautiful parks, great pubs and independent coffee places on the high street, and there are plenty of Victorian terraced houses or flats within easy walk of the high street. There are also plenty of play groups, and baby stuff (yoga/massage/sensory/music etc) going on. PP is correct though that renting is likely to be more expensive here than in Kings Heath or Moseley.
There are excellent primary schools in Harborne, but they are oversubscribed so catchment areas are small, and it is perfectly possible to live in Harborne but not get a place at any of its schools. So if you are likely to still be here when that becomes an issue, you'd be wise to take care in choosing where exactly you live relative to the schools.
Theoretically, all three of those areas are within 10/15 minutes bus ride from the centre of Birmingham, but that is only outside rush hour. The journey time will be very much longer at peak times, and that can get unbelievably frustrating. The buses from .Harborne stop very close to the Children's hospital, not sure about the Kings Heath and Moseley buses. Personally, I cycle in to town from
Harborne to avoid the dreadful traffic, and would do the same from Kings Heath or Mosley if I lived there.
You could also have a look at Bournville. That is a very leafy area - prettier than all the other areas mentioned above. It has an excellent primary school, I believe, and good community facilities. No pubs or bars whatsoever though, but better public transport. It has a railway station so you can get into town on the train, thus avoiding the traffic.
There are independent small shops including butchers and bakers, but I don't think there is a supermarket in Bournville itself (which probably contributes to the relative beauty of the place). There are supermarkets in neighbouring areas such as Selly Oak, and you could easily live in Bournville and be within walking distance or an easy bus ride of a supermarket.
Good luck!