I am very new to illustrator. I want to cutout a stroke of a circle (fill color is turned off) using another object and then color the segments like the below steps:
Expand your circle stroke (so that it becomes a path) by going to Object → Expand and only choose Stroke
You can now use the Shape Builder Tool or Pathfinder to break up the shape.
Pathfinder (Window → Pathfinder or Shift+Ctrl+9);
Select all and click on Minus Front
Shape Builder* (Shift+M);
Select all and then Alt+Click+Drag on the areas you want to remove from your shape
Now color your shapes and apply shadows how you'd like
*Shape Builder can be a better option when dealing with very complex shapes, as pathfinder won't usually perform well. In your case, Pathfinder is very suitable (as it is very simple).