We are a family of four (2 young kids), and are planning on travelling from Hong Kong to Guangzhou to Shanghai in late May/early June. Could someone recommend the most convenient way of getting to and from these cities? Appreciate any input, thanks.

haha i have done exactly the same as you did.. Left HK on 20th january.. that time was the most hectic in buying tickects due to spring festival..
HK -> GZ Train best option.
GZ - SH.. Train about 20 hours, since you have 2 young kids i would suggest a plane or a cabin for your family, dont go the hard/soft seat option. my opion would to book a flight.
I was standing for 20 hours with 3-4 smokers infront of me.