There are TONS of apps for notes. It depends on your preferences. Evernote will sync across all platforms and devices, including your laptop/desktop if you're using Mac's. (not sure if it does the same with pc's) You can take photos to use with Evernote as well. They update frequently-but not too frequently.....just what's needed. You can organize it however you like.
i use one called "Catch". Unfortunaely it's not in the app store, web-based, but it's got a beautiful interface and very clean. They have what they call "spaces". I don't know when/if it will be in the app store. It has disadvantages in this regard.
You can also use the "Reminders" app on your phone. I don't know if you can categorize with this or not, plus when you check off a reminder, it disappears, so it might not be what you're looking for.
Don't have an answer to your "notes" question as i don't use it....sorry. i'm sure someone will pop in to help! take care!