Something like that is going to be most easily accomplished with a vector application, such as Illustrator, Inkscape or CorelDRAW.
Learn to use the Pen tool, and possibly whatever Boolean tools are available. (In Illustrator, they're called Pathfinder tools.) A Boolean tool allows you to do things like creating a new shape based on the overlap of two existing shapes: Adding them together, keeping only the intersection, keeping everything except the intersection, etc.
Learn how to apply gradients and play with opacity/transparency settings. Very small glows are useful for getting that luminous appearance.