At steamlabs, we've been helping kids to make robots at our makerspaces for over 8 years in our summer camps, weekend and after school programs. One of the things kids find most challenging is mechanisms. It's hard to anchor a robot arm's pivot points securely and transfer the motion of a motor into that pivot point.
To address this, we've developed an easy to use cardboard robot construction kit. It consists of standardized laser cut cardboard and wooden parts. Interlocking rectangles and boxes form the basic structure. Holes allow plastic straw and chopstick axles to easily turn. Wooden levers, gears and pulleys are easily positioned in the holes. Elastic band drives can be added for simple motion, and perforated cut outs make it obvious where to add motors and servos. Linkage bars allow easy transmission of motion into any other parts. You can order the parts from us for very affordable prices. Everything is open source, so if you have access to a laser cutter, we provide the files for you to cut them yourself!
A great starting point for robotics is to make something with servos or other electronics with manual controls.