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Usually, I'll turn off data at certain times of the day, but once I go into the Notes app to read notes I've jotted down, it prompts me to turn on data, why would this app require it?

Your Notes sync over iCloud, and since Notes are kbs in size, it does it using data.

You should still be able to read your notes though, is it not letting you?
 
It's letting me, but when I have data off, it keeps alerting me to go to settings to turn data on. I was hoping I can turn that off as it's annoying.
You can perhaps turn notes off from any account you might have (iCloud, etc.) on the phone and perhaps that will stop prompting you to turn it on for sync purposes.
 
My notes sync with my mac, so when I write something down on my iphone it appears on my mac.
 
I just wish the syncing was better.

I end up with duplicate notes when I edit them a lot between my iPhone, Mac and iPad. Very annoying because they generally look fine on my Mac but have multiples on my iOS device of the same notes however slightly different. I find myself fixing something that should "just work" quite often.
 
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