@Callum-Jones
This demonstration uses the ULN2003A, which is a cheaper version of the ULN2803, but is the same as far as electrical connections and such.
I wrote this in simple terms for clarity so anyone can understand it, hopefully.
In the Fritzing diagram below it shows the connections to the Onion Expansion and your external power source. This would be the same for whatever you are trying to power, wether it be an LED or a relay.
The ULN2003/ULN2803 sink the Ground to your part. So if you are trying to run LEDs or other directional power parts you have to have the Negative line(Cathode) of your part connected to the output of the chip and the positive leg of your part connected to the positive side of your power source.
When I hooked it up I just used the GPIO Tool in the Onion Console to turn on or off each color as I wanted. It worked flawlessly. My boy, only 4 years old, wanted to turn the colors on and off too.... lol
I hope this helps. Feel free to message me if you have any questions.