There is no actual rule for this, so we can say that both are interchangeable most of the time. But ‘saya’ is considered more acceptable in a formal situation, especially in writing; even though you can still use ‘aku’ when you want to talk to your professional business client or your bosses at work. 

So when you talk to your boss and you feel like you are quite close with her/him, you may probably speak like this:

Boleh nggak aku ambil cuti besok? / Can I take a leave tomorrow?

But if you think that your relationship is strictly professional, you may settle with sentence like this:

Saya ingin minta izin cuti besok. / I would like to ask for your permission to take a leave tomorrow.

There is also no rule saying that you cannot mix both saya and aku in one sentence or in one conversation. But it will sound a bit weird. Example:

Aku selalu minum teh tanpa gula, karena saya punya risiko diabetes. / I always drink tea with no sugar, because I am at risk of diabetes.

Also, if you use saya as a personal pronoun, you should use saya too as a possessive pronoun. And if you use aku as a personal pronoun, then you should use –ku or aku as a possessive pronoun. 

So for this sentence: I am going to pick up my kid this afternoon.

Saya akan menjemput anakku nanti siang. → not good

Saya akan menjemput anak saya nanti siang. → good 

Aku akan menjemput anakku nanti siang. → good

Just be consistent in each situation.