hikkosita ato, mazu suidou ga tukaeru ka douka kakuninsimasita.

Because あと in this pattern usually follows the plain past form when you mean after doing something.

• 引っ越したあと = after moving
• 食べたあと = after eating
• 帰ったあと = after going home

This contrasts with まえ:

• 引っ越すまえ = before moving

So a useful pattern is:

• Verb past + あと = after doing X
• Verb dictionary form + まえ = before doing X

Here, あと means after and works like a noun.

So 引っ越したあと is literally something like the time after having moved.

That whole chunk sets the time for the rest of the sentence:

• 引っ越したあと、…確認しました。
• After moving, ... I checked.

You may also see:

• 引っ越したあとで

This means basically the same thing, with a slightly more explicitly connective feel.

まず means first or first of all.

It shows sequence or priority:

• まず水道が使えるかどうか確認しました。
• First, I checked whether the water was usable.

It suggests this was the first thing done after moving, probably before checking other things like electricity, gas, internet, and so on.

水道 means water supply, running water, or plumbing/water service.

That is different from 水, which just means water as a substance.

So:

• 水 = water
• 水道 = the water service / tap water system

In this sentence, the speaker is not checking whether water exists in general. They are checking whether the water service in the new place is working.

This is a very common learner question.

The base verb is:

• 使う = to use

So normally:

• 水道を使う = use the water supply

But in the potential form:

• 使える = can use / is usable

Japanese often marks the thing that can be used with が, especially in this kind of sentence.

So:

• 水道が使える = the water supply is usable / I can use the water

This is very natural. In English it may feel like water supply should be the object, but in Japanese potential expressions often behave differently.

Also, here 水道が使える sounds almost like describing a state:

• the water works
• the water is available

That is why が feels especially natural.

使える is the potential form of 使う.

• 使う = use
• 使える = can use / be able to use

In this sentence, it does not just mean personal ability, like I know how to use it. It means something more like:

• is available
• is working
• can be used right now

So 水道が使える is very naturally understood as:

• the water is on
• the plumbing works
• the water service is usable

かどうか means whether or not.

It turns the clause before it into an embedded yes/no question:

• 水道が使えるかどうか = whether the water supply can be used

Then that whole clause becomes the thing being checked:

• 水道が使えるかどうか確認しました。
• I checked whether the water worked.

This pattern is very common:

• 行けるかどうか分からない。
  I don’t know whether I can go.
• 本当かどうか確かめる。
  Check whether it is true.

Yes, sometimes か alone is possible:

• 水道が使えるか確認しました。

This also means I checked whether the water could be used.

The difference is:

• かどうか more clearly means whether or not
• か can sound a little more compact

In many situations, both are natural. かどうか often feels slightly more explicit.

Because clauses ending in か or かどうか often connect directly to verbs like:

• 確認する = confirm/check
• 知る = know
• 分かる = understand/know
• 覚えている = remember

So this is natural:

• 水道が使えるかどうか確認しました。

You may sometimes see を with other nominalized expressions, but with か / かどうか, leaving out を is completely standard.

In other words, the whole clause functions as the content of what was checked, even without an explicit を.

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

From context, the most natural subject is:

• I checked
• or possibly we checked

Japanese often leaves out subjects when they are obvious from context. So a full version could be something like:

• 私は引っ越したあと、まず水道が使えるかどうか確認しました。

But in natural Japanese, 私は is often unnecessary.

In normal Japanese writing, you usually would not put spaces between the words. It would normally look like this:

• 引っ越したあと、まず水道が使えるかどうか確認しました。

The spaced version is often used for learners to make the parts easier to see.

A rough breakdown is:

• 引っ越したあと = after moving
• まず = first
• 水道が使えるかどうか = whether the water supply was usable
• 確認しました = checked / confirmed

So the spaces are just a study aid, not standard Japanese formatting.