As a child a favourite toy was my Spirograph with which I could generate endless pretty curves!
The Nrich problem ‘Making Maths: Planet Paths‘ challenges students to draw some planet paths using a Spirograph. In case there is no Spirograph to hand they give instructions for making a simple one.
Alternatively try an online applet. This from wordsmith.org allows you to have the moving circle either inside or outside the fixed circle. You can alter various parameters and experiment.
From mathplayground.com comes this very clear and easy to use applet (the moving circle is outside the fixed circle).
For GeoGebra fans there are various applets available, including this which allows colour changes.