Need more powerful tools?
The simplest thing is to share the PDF to Apple’s Notes program. Here are the steps to bring a grid into Apple Notes:
Because you can use any drawing app that can edit a PDF, choosing a full-fledged graphic design program will give you more capabilities than Notes. One app we like and use is Graphic for iPad. It has an amazing amount of power for not much money and also works with an Apple Pencil. It has features like 'snapping to grid lines' that makes adding shapes easier. Also with Graphic for iPad, you can add lines, notes or even modify the shapes originally created in our app, for example, and change the line color or width.
Coloring a grid using Graphic for iPad
[Also available for the Apple Mac]
PS. We have no association with Graphic other than we use it.
When you export the PDF to Graphic, have Graphic “Import as Vector Objects”. Now you can add lines/text on top, and you have access to all the lines in the drawing. So, for example, in the image below, you can grab a section of the grid and change the colors of the lines in that section. One can also change line weights - or get crazy and add arrows or dashes. One can even grab and move things around on the page.
The other thing to point out is a “Smart Guides” feature the program has that will help you snap to the grid lines on the paper rather than Graphic’s internal grid points - which probably will not match up. This will make it easier to draw on your own lines - particularly with an Apple pencil.