When creating your laser- cut design, you should ensure that only the vector line information NEEDED to make your design is present in your file. You never want to try and hide cut lines or vector engraving lines behind fills - they will be made by the laser-cutter whether visible in 'Preview' mode or not.
The good news is that it's simple to create complex shapes in Illustrator using a set of buttons in the Pathfinder window that will make your laser-cut design creation process much easier. You can find the Pathfinder window by using the Window > Pathfinder command - and the names of each of the buttons in the window can be found by hovering the mouse pointer over them.
Below are guides to using two of these buttons, and we recommend you experiment with the rest.
Using Illustrator's Pathfinder > Minus Front button - step by step
1. Draw an initial large shape, that provides the outer edge of the final shape you want to make
2. Draw a second shape, with the area you want to remove from the first shape overlapping
3. Select both shapes
4. Use the 'Minus Front' button in the Pathfinder Window
5. Voila! The second shape has been removed from the first - creating an 80s video game character
Using Illustrator's Pathfinder > Unite button - step by step
1. Create more than one shape on your template that you want to combine - all with the same fill/stroke
2. Adjust the placement of each of these shapes so that they overlap with each other
3. Select all the shapes
4. Use the 'Unite' button in the Pathfinder window
5. Voila! All of your shapes have now been combined into a single part for cutting out
This series of tools are invaluable for creating complex designs for laser-cutting, and you can use them in a whole range of combinations. We'd recommend having a play with each of them, and finding out how they can help with what you'd like to make.