I guess that cylinder is the heating element?
I don't see a relay in your pictures.
I don't know what those other boards are and if you don't have a schematic or spec you probably can't use them. (If you need a display, it's probably best to start-over and build something you understand (or something that can be understood.)
I just don't want to kill myself trying to power this coil.
Attached are pics of the coil and the circuit it came from. hopefully that gives enough info.
Well... Power line voltages can kill you so if you don't feel like you know enough to be safe you probably shouldn't be messing with dangerous voltages. It won't always kill you... I've been shocked by 120VAC and 240VAC a few times, but it is dangerous.
Have you ever changed a light switch in your house, or anything like that? Do you understand how a light switch works? ...Changing a light switch (or removing the switch and installing a dimmer) is generally safe because you should turn-off the power first... Then, the worst thing that can happen if you wire it wrong is that the circuit-breaker blows when you turn the power back on.
Messing around with the Arduino and AC power is more dangerous because you are "mixing" low-voltage & high voltage circuits, and possibly touching the Arduino, or plugging the Arduino into a computer, etc., and "bad things" can happen if the high voltage gets connected to the low-voltage circuitry.
If you feel you can wire-up an AC plug to your heating element, something like [u]this[/u] will keep everything else safe.
Do you have a multimeter? With a multi-meter you can measure the resistance between the AC circuit and the low-voltage circuits (with power disconnected!) to make sure there is no electrical connection (an "open" or infinite resistance).
If the circuit is isolated and insulated do you can't touch the hot wires, you are reasonably safe. (You aren't necessarily safe from a short or other wiring error than can cause overheating and fire, but you are reasonably safe from electrocution.)
A relay provides electrical isolation between the low-voltage Arduino and the power line. (It's still a good idea to check the isolation with a multimeter to make sure you've wired it correctly and that you truly have isolation.)
With the Arduino, you generally need a relay driver circuit because the Arduino can't directly provide enough current or voltage to activate a relay coil. But, "relay boards" with the relay, built-in driver, and screw terminals are common. Alternatively, solid state relays that can be driven directly by the Arduino are easy to find. (But, solid state relays are more expensive.)
. The original thought was to make a circuit similar to the popular fan activated by the temp sensor (tons of vids on youtube).
The circuit for controlling an AC fan would be (could be) the same. You've got a temperature sensor as input, and a relay-controlled fan or heater on the output. You just need to reverse the logic in your program a bit.
And, I would not recommend blindly following some random person's YouTube video... Try to understand how the circuit works... Study some basic electronics... Check out some similar circuits/schematics and try to understand how it works.
Same thing with the software. Try to understand how the software works. Reading the temperature might be a little confusing for a beginner (you can use a thermistor or a solid state temperature sensor) but the basic logic that says, "turn-on the heat if it's too cold and turn-off the heat if it's too hot" can't get much simpler! In fact, a "basic" non-programmable thermostat in your home, in your oven, or in your refrigerator doesn't have a microcontroller, doesn't have any software, and doesn't have any memory.
