How should I start an electronics project?
Designing something like this will take a while, and it's too much to really do through a site like this, but I can at least tell you how I would set about it:


Break the task into chunks, and do each in turn, and
Start with the important bits - in this case, it's going to be all about the interface.


There's a whole lot of mechanical stuff to sort out too, but since this is the electrical engineering stack, I'll skip that.  Then I'd break the electronics into several tasks:


The buttons. They need bright illumination, a switch so they can tell they've been hit, and they need to be robust.
Displays: one or two largeish 3-digit, 7-segment displays. These you can just buy, it's a case of looking at what's available and what you want or can afford.
Other lights: like the displays, build a shopping list.
Sound: I'm sure that's something to look at too.
Once you have a good idea what you want for an interface, you need to work out how to do the brains of it. A microcontroller or a microprocessor. Exactly what type will depend on what you have in the points above.  You might find, for example, that you can do everything with an Arduino or similar except the sound, in which case you might want to go back and change what you want there. Or you might decide that fully detailed stereo is important and get an ARM board instead.
Power. This is usually good to do last, once you know what you need to power, though again you might end up revisiting the above points to save some money or simplify things.


Then, to take an example, lets look at point 1. We can break that down again into even smaller bits. You'll need a robust frame and a semi-transparent dome, but that's not really electronics, so let's skip that for now.  It also needs a light that can be switched on and off, and a switch to tell when it's pressed. 

Lets start with the light. Have a look at some lightbulbs and how bright they are. Look up their light output online, and work out how much light you need. Pick a technology - probably LED. Pick a specific light. Work out what voltage and current it needs. Check if that is too much for a microcontroller, probably it is, so you'll need a MOSFET or similar to switch it on and off, so pick one. Work out how to drive it from a generic low voltage logic signal (you'll fill in the details when you pick a microcontroller). Choose the other parts you need to make a complete circuit, such as resistors etc. Finally, check the heat dissipated by each part, and choose heatsinks as necessary.  Then move on to the next task on the list.

Good luck!