DIY Coupling: A coupler for the motor and lead screw (from Marc K.)
This creative coupling solution comes from another generous user of the site. He has devised a way to couple the motor to the leadscrew with standard aluminum tubing. Marc has graciously allowed me to show you this creative idea.

Step 7: Z-Axis Transmission Nut
This tutorial is dated, if you are considering a CNC for your personal use, we would highly recommend purchasing a kit that is very stable from our wide range of machines available. Instead of using allthread for the lead screw and a standard nut, consider using a 5 start lead screw and anti-backlash nut. It will keep you from pulling out your hair by running much smoother and faster. We will install the method that provides transmission to the z-axis. The transmission consists of a lead screw and a nut. Very simply, the motor will turn the screw, and the nut is fastened to the z-axis assembly, moving the assembly as the motor spins. That is to say, when the motor turns clockwise, the nut will travel up, and if the motor turns counter-clockwise, the nut will travel down.

Step 9: Z-Axis Lead Screw and X-Axis Aluminum Angles
This tutorial is dated, if you are considering a CNC for your personal use, we would highly recommend purchasing a kit that is very stable from our wide range of machines available. Instead of using allthread for the lead screw and a standard nut, consider using a 5 start lead screw and anti-backlash nut. It will keep you from pulling out your hair by running much smoother and faster. Lead screws... hmmm... ok, let's talk concepts. The lead screw is the main component for the transmission of power in a linear form. Yes, a little like the transmission in the car, but much simpler. The screw yields a ratio of force to the linear motion, but conversely affects the speed and resolution of linear motion. To explain in more simpler terms, take a regular screw, and a nut. Hold the nut in one hand but don't turn it. Hold the screw in the other hand and screw it in the nut. Your hand is applying a force to the screw in a circular motion, hence the motor turning the shaft, and if the screw is turning at a stationary point in space, it will pull the nut toward the hand turning the screw. If the turning motion of the screw is reversed, the nut is pushed away from the hand turning the screw.