It depends … Sometimes I work on my iPad, so saving is easy. The (loose) paper sheets are scanned on a flatbed scanner. I don’t do much cleaning up at the moment. If I am in a hurry, I just take a photograph with my iPhone.
If you are familiar with photo editing software you can go further with cleaning up and vectorizing. I personally am not willing to invest into Photoshop etc. at the moment; I have other software which will do the same stuff (if I have the time to learn how to use them). But that would be the usual way to go to clean up an image.
There are apps for iOS though that allow loading of a jpg/png image and converting them into a vector graphic.
Many ways to explore!