My father said he would clear a day to take me to the zoo.
Literal
Father [topic-は] me [object-を] zoo [to-に] take [purpose-ために] day [object-を] keep-free [volitional-おこう] [quotative-と] said.
日をあけておこう uses ~ておく (prepare in advance) in the volitional form (おこう) — 'I'll keep a day free.' The whole phrase is embedded in indirect speech with と言った.