What are some grilling tools that everyone should have

Metal
Advantages: durable, no risk of absorbing flavours
Disadvantages: may scratch certain cookware

Wood
Advantages: doesn't scratch, looks nice
Disadvantages: can wear down over time, can sometimes absorb flavours

Plastic
Doesn't find a place in my kitchen other than a rubber spatula for baking.

Silicone
You didn't mention this, but I like it, because it works like rubber but can withstand high temps (brushes, spatulas)

I think the answer really depends on what you cook. A good basic set might consist of a wooden spoon, a spatula that can handle heat, a good ladle, and maybe a set of tongs or chopsticks. You'll quickly learn what you wish you had to make it a little easier.

If you have access to a well ventilated area and don't mind using charcoal, shichirin are my absolute favorite style of small grills. They also come in various shapes and sizes. You really should use them outside, though, since they release carbon monoxide and their temperatures of 1200°C+ can create a lot of smoke if you are grilling something fatty.

In my experience, small electric grills don't really get hot enough to do anything useful culinarily that a regular range/cooktop couldn't do.

If you must use gas, you could try getting a propane or natural gas camping stove, some of which come with grills.

It won't be as powerful, but there are also some small grills on the market (also see this one) that work off of convenient butane canisters.

Finally, if you already have a gas stove, there is a type of handheld grill that is popular in Japan. It has a metal grill grate on top of a ceramic diffuser. It looks like this:

The ceramic diffuser is the white part below. You hold it over your gas stove and the ceramic evenly distributes the heat across the grill grate. Unfortunately I don't remember what these things are named. Does anyone know?