How can I get from Hung Hom train station to Hong Kong International Airport

You can take bus no. A21 near Hung Hom station to HKIA directly. The bus fare is 33HKD. There are luggage racks onboard for you to put your luggage. So it is generally convenient to travel by that.

If your kids are very small and you have many luggages, I would suggest you take a taxi to go instead. The cab fare is around 235HKD for your reference.