maki to tyakkazai wo tukau.

It is read:

まき と ちゃっかざい を つかう。

Breakdown:

• 薪 = まき
• 着火剤 = ちゃっかざい
• 使う = つかう

Here, と means and.

So 薪と着火剤 means firewood and fire starter.

This use of と connects nouns in a complete list.

Because 薪と着火剤 is being treated as one combined object.

Japanese often puts the particle only once, after the whole noun phrase:

• 薪と着火剤を使う
• literally: use [firewood and fire starter]

You do not normally say:

• 薪をと着火剤を使う ✘

A single を at the end is the normal pattern.

Because 使う is a transitive verb, meaning it takes a direct object.

The thing being used is marked by を:

• 道具を使う = use a tool
• お金を使う = use money
• 薪と着火剤を使う = use firewood and fire starter

So を marks what is being used.

Yes, と can also mean with, so this is a very common point of confusion.

Here it means and because it is simply joining two nouns into a list:

• 薪と着火剤を使う = use firewood and fire starter

Compare that with:

• 友だちと行く = go with a friend

So と can mean different things depending on the sentence structure. In your sentence, it is the listing and particle.

着火剤 literally means something like ignition agent.

In natural English, it is usually:

• fire starter
• firelighter
• ignition aid

It is something used to help start a fire.

It is not exactly the same as kindling.
If you specifically mean small sticks or material used to help catch fire, Japanese often uses 焚き付け（たきつけ）.

So:

• 薪 = firewood
• 着火剤 = fire starter / firelighter

The subject is omitted, which is very normal in Japanese.

Depending on context, this could mean:

• I use firewood and fire starter
• We use firewood and fire starter
• Use firewood and fire starter
• One uses firewood and fire starter

Japanese often leaves out the subject when it is obvious from context.

使う is the plain form, also called the dictionary form.

It is commonly used in:

• casual speech
• written instructions
• labels or notes
• neutral statements

The polite version would be:

• 薪と着火剤を使います。

Both are grammatically correct. The difference is politeness and style, not basic meaning.

Good question. Both can connect nouns, but they are not the same.

• と = an explicit, complete list
• や = examples, a non-exhaustive list

So:

• 薪と着火剤を使う = use firewood and fire starter
• 薪や着火剤を使う = use things like firewood and fire starter

With や, the listener understands that other similar things may also be included.

Yes, you can.

• 薪と着火剤を使う
• 着火剤と薪を使う

Both are grammatical and mean essentially the same thing.

Changing the order may slightly change emphasis, because the first item can feel a little more prominent, but the basic meaning stays the same.