Trigger 12v

The way I do this is with an optocoupler, which means you don't need to worry about creating ground loops even for your control circuitry that should be isolated from the audio circuitry, but still grounded.

You can drive a big mains relay using a little FET (AO3400) or BJT (BC548/2N2222), which can easily sense when the optoisolator is enabled. Put a normal soft-start circuit after the mains relay.

You can even buy opto-isolated relay modules on eBay which contain everything you need to switch mains power from a 12V signal. They tend to have cheap/fake/songle relays though which are not good for as much current or voltage as they claim.

Here's a schematic showing how the opto-isolator/transistor/relay combo works; look at the top-left section. This is in the context of a preamp which lives in the same box as a power-amp, so the K1 relay is controlling power to the main power-amp supplies as well as the little linear supply that is on the same PCB as this.

If you want to be fancy, you can use a microcontroller to detect the remote trigger signal and drive the relay. Connect some temp sensors, do PID fan-speed control based on temperature, and disable the power if there is an overheat or fan failure. Example arduino code.
 
The absolute most braindead way is to just connect the remote-in socket to your relay coil. That means you don't need your own 12V supply inside your amp, but it does mean that you are depending on the master/source to supply enough current for the relay. Most will not. And you don't want to push an inductive spike back into the master when the relay shuts down, so it's really not a good idea.