There is a huge number of ways to save data in iOS.
Instead of saving the notes to the device itself, you might consider saving it to a back end server instead. I would recommend taking the time to get a basic familiarity with Node.js + MongoDB. You can build out your own backend. This way, a user with a single account can see all of their data on all of their devices.
However, if you definitely want the notes to be saved locally, there's a few technologies I would recommend. CoreData can be powerful but it is extremely quirky and difficult to work with. Realm is a newer database framework which I have never used in a production app but I have been experimenting with lately and I like it quite a bit.