Two separate things.
1) You can save stuff on_pause() [ see any Python tutorial for writing files], but you don't need to.
p4a will automatically save/resume any class variable (self.whatever) state.
2) Either
a) Save the information used to create the drawing instruction in a class variable. So for example when you define a new_line used by a Line() canvas instruction add it to a save list self.lines.append(new_line). Then create the canvas instruction from the saved data. Obviously there is other data to save, you are probably going to create a list of dict.
b) Consider using InstructionGroup to create a set of instruction and gave the group in class variable.
I don't have an example.