everything stays on when keys out

[Niall]

The remote wire is the blue one...but that's for powering an amp so I guess that isn't what you meant lol.

But what you need to do now is swap the red and (other) yellow wire over so it gets a switched live with the ignition.

This. They should of put bullet connectors on the red and yellow wires so you can swap them. What make of stereo did the loom come with out of interest?

Cut the yellow and red and swap them around with bullet connectors so you have red leading to yellow and yellow leading to red (the yellow on its own BTW. Not the one you have just cut to stop the back feed)