That's probably a good place to start. You can replace the current 7 or 8 leds with a shift register by plugging it in to the breadboard and moving the led wires from the Arduino to the shift register outputs. Then you need to modify the arduino code to transmit the values that drive the leds to the shift register (the arduino tutorial should help with that). That reduces the number of pins you need on the Arduino from 7 or 8 to 3 for the shift register. Once that is working you can add a second shift register to get even more leds (again you will need to modify the Arduino code to drive them).
Peter