For drawing directly on a surface, turn off "Straight Snap" in the bottom toolbar so that it does not kick in and also go to the View section on the side pane and uncheck "Display hidden lines", so that you won't snap to the back edges of the shape either. You may also want to go to the "Object Snap" button in the bottom toolbar and push the little arrow that pops up above it and uncheck "Axis" so that you won't snap to the x and y grid axis lines either. You do want Object Snap turned on so that you will snap onto the surface though. With things set up in that way you can then use Draw curve > Freeform > "Through points" and place points on the surface to form a curve like this:



That's another way to get some curves to work with - note that the curve drawn in this way will only be exactly on the surface at each of those picked points, so when you generate a sweep or something from it you will probably want to have the sweep be generated as some piece that sticks out a little bit so that you can be sure it will intersect the outer shape fully.

- Michael