Wireless Button Press to Have Program Run for Set Amount of Time

Completely new to Arduino and am looking for some advice on how you would go about what I'm trying to accomplish.

I have a 24V rotating siren style light (no sound) and an announcement speaker that accepts SD cards. What I want to happen is have an arcade style dome button that when pressed sends a wireless signal a few feet to the siren and speaker that runs for 5 seconds and then shuts off and can't be pressed again for 10 seconds.

I already have the components wired up (less the arduino) and I can get it all to work by holding the button (hardwired, not wireless) down, but I need the button press to cause it to run for a set amount of time without holding down the button.

That is the quick breakdown of what I'm trying to accomplish, and like I said I know next to nothing about arduinos so if you have any suggestions that could potentially save me some time or frustration I'd appreciate it. I haven't found any similar project that I could use as a starting point. Let me know if I don't have enough details about something. Any ideas or suggestions?

Treat this as two separate problems. Learn how to make the device run for an amount of time. The demo Several Things at a Time illustrates the use of millis() to manage timing.

Separately learn how to send data wirelessly. My experience is with nRF24L01+ transceivers which are cheap and work well. But they may be overkill for what you want. You can get simple key-fob transmitters that may be sufficient.

When you can do the two parts separately you can then amalgamate them into a single project.

Simple nRF24L01+ Tutorial