Reddit mentions of ELEGOO 8 Channel DC 5V Relay Module with Optocoupler for Arduino UNO R3 MEGA 2560 1280 DSP ARM PIC AVR STM32 Raspberry Pi

I'm not really pro-grade in this kind of stuff, however, if I were to go after something like this, an Arduino seems almost perfect. For anything you need to control, I'd use this type of relay designed to plug straight into Arduino IO pins and as far as the inputs, you could probably use relays wired up to a simple 5v source so you isolate the power within the pinball machine from clean 5v power for your Arduino.

As far as the display, instead of an LCD, I would probably chase some larger 8-segment displays, easier to see and much easier to control. One option would be to run 2 Arduino systems with one dedicated to the scoreboard and one dedicated to the automation.

ok, ramble off, but this sounds like a project I'd like to take on now...

There's not much to it, depending on the relay you have.

Relay's have two parts, a coil and one or more sets of contacts. The coil is what controls the contacts. The coil is an electromagnet, and the contacts are reed switches. When the coil is energized, it causes the reed switches to engage. So by using the contacts as part of a circuit, you can make or break the circuit by engaging the relay.

Relay coils by themselves pull too much current to safely run one with a raspberry pi, so you'll most likely need a transistor to control the relay (a transistor can act more or less like a tiny relay with no moving parts).

If you've bought a pre-packaged relay board (such as one like this) then they've taken care of the transistor for you and all you have to do is hook VCC to the pi's 5V pin, the ground to the pi's ground, and the input you want to control on the board to the output on the raspberry pi.

There are plenty of tutorials out there that show you how to use python to control the pins.

On these particular relays, there are three terminals for the contacts, NO (normally open), common, and NC (normally closed). To switch a circuit you connect one wire to common (usually the middle terminal) and the other wire to either NO or NC depending on if you want the circuit to be On (NC) or Off (NO) when the coil is de-energized. Don't connect a wire to NO and the other to NC and expect anything to happen, they're not connected together. One wire must go to Common for there to be a complete circuit. You are free to connect to all three and switch your common line between two different circuits (one for NO and one for NC)

You dont. At least not directly without killing the board. That said there are ready to go relay boards you can order if you want a simple solution, or go read the other comment about using a transistor, he/she is dead on on how to do it.

That said, a transistor is kinda like a relay. Perhaps that's all you need if you are running a small device like a few LED's or something else small.

Take a look at using a contactor with a smaller relay to act as as a "bridge" to a MCU.

In other words (assuming you're working with two 120V "hot" legs and a neutral - verify this!):

https://imgur.com/fEHNtIr

You might want to use a 5V relay board to make it easier to wire.

Just make sure that you mount all the components in a safe enclosure (~$25 on Amazon).