Go to a Mother and Baby Clinic (I forget the official name) and you can have all the checkups and jabs done for free. They'll arrange for baby health checkups at the Specialist Clinic for you. It's all very well organised. However, no home visits.
Also, in the HK private health system, you normally see a GP, who may recommend you to specialist paediatricians in the private sector if required. The public sector works in a similar manner.