A common way to draw circles is the draw them as line segments. more segments for larger circles, with some minimum number of segments so they look good. Since you are moving a pen it isn't as critical as compared to a machining operation, though delays at intersections can cause some bleed with a felt tip pen. I have an old commercial plotter and I use fine line Sharpies and some fine line ball point pens. You will need to play with them to get the best parameters. Speed is important. Some pens with write faster than others.
for motion - you need to be able to work in 2 coordinate systems - things like circles you need to be able to make your calculations as if the circle is centered at (X,Y)(0,0) and then be able to convert them to your "world" or drawing coordinates. It can all be done in incremental mode, but it is easier to visualize in absolute.
Incremental mode is where every move just references off its start. Absolute mode every position is relative to some (0,0) position on the machine/drawing.
An incremental square 2" from the corner looks like this move 2,2, move 1,0, move 0,1, move -1,0, move 0,-1
same for absolute - Move 2,2, move 3,2, move 3,3, move 2,3, move 2,2.