We had a full day on Boxing Day of exploring the Botanic Gardens with our young kids (aged 8 and 5). We all enjoyed the shuttle around the gardens with knowledgeable guide who pointed out so many interesting things, punting on the lake (we didn't book ahead so had to wait 2 hours on such a busy day), and exploring the gardens on foot. A big walking day and it was quite hot, but enough shady places to keep cool.
I recommend taking your own food, snacks and water as the cafe on site is crazy expensive. We took everything with us, but paid top dollar for an ice cream treat for the kids. We didn't mind as the money clearly goes to the upkeep of the gardens and grounds.