Here is how it was done in 1810...By a mechanical wiz, not a beginner - now show us how you'd do it with a few motors and a controller
And I don't think that you'd put that old mechanical doll in the "simple" category?
Simple it was not. Just imagine what it took to program it by cutting brass cams for each degree of freedom... Amazing is what I would call it.
Now, back to the original problem - is this robot supposed to write random phrases or just a couple pre-programed things?
One of the tough parts would be the math - first, mathematically describing the path for the tip of the arm and then sorting out all of the joint angles as a function of time to get it to trace the desired path.
One way around the math would be to build the arm with position sensors and then just move the arm by hand along the desired paths and record the angles. Then, the recorded values could be used as the target positions for your arm controller (a simple PI controller is likely to be adequate if you didn't try to move the arm too fast). Create tables of time / position pairs to describe the paths for the arm.
If the arm was built with servos - that makes it more difficult to record the "training" movements because all the feedback is inside the servo's themselves. So, you would have to open up the servos and run wires back from the potentiometers to your micro-controller to record the servo positions as you train it. (And translate from the voltage from the pot to the pulsewidth sent to the servo to get that position - another training exercise where you command various pulsewidths and record the potentiometer output.)
Getting a robot to generate nice cursive writing was a challenge in 1810, and can be a challenge today!