Thank you everyone for all your help! I thought I should post the finished product in case someone in the future has the same problem. You were all right, I'd counted in from the right on my board, not realising there were 18 connectors instead of 16 like on non-RGB boards, so my wiring was all kaput.
A bit frustrating because the information, for a newbie at least, isn't that clear and the tutorial the product page sends you to just works from a model that only has 16 pins instead of 18. It DOES mention it briefly on the product page, but I missed that the first time around.
Eventually figured it out though (well, my wife did actually, hence the insisted upon message on the finished version).
I've uploaded an image of the finished version.
I used this tutorial for the wiring and coding: https://learn.adafruit.com/pages/243/elements/2973928/download
This one says that you don't need resistors because they are built into the board, but on the board's product page it says to use resistors or you can burn out the RBG colours.
Additionally, on the schematic for wiring above, it has a red jumper cable going from pin 15 to a positive rail on the side. It was only when I removed this that the board lit up and showed the message, albeit, quite dimly.
When I realised on this datasheet: https://cdn.sparkfun.com/datasheets/LCD/Monochrome/ADM1602K-FS-RGB-FBW.pdf
that pin 15, labelled K, is ground, I put a black jumper cable into a negative pin on the negative rail on the side and it lit up like Christmas.
Don't know if I'm doing this right, or I'm going to blow stuff up, but for the moment, it's working. 
Thanks again for all of your help!