Ég hefði átt að taka regnfrakka með mér, en í staðinn fór ég bara í mátunarklefann og keypti annan jakka.

It is a common Icelandic way to express should have done.

• eiga að + infinitive = to be supposed to / should
• hefði átt að + infinitive = should have + past participle in English sense, meaning something was advisable or expected, but did not happen

So:

• ég á að taka = I should take
• ég átti að taka = I was supposed to take
• ég hefði átt að taka = I should have taken

Here, hefði is the past subjunctive of hafa, and átt is the past participle of eiga in this idiomatic construction.

Because regnfrakka is the accusative singular, and it is the direct object of taka.

The dictionary form is:

• regnfrakki = raincoat (nominative singular)

But after a verb like taka when the noun is the thing being taken, Icelandic uses the accusative:

• taka regnfrakka = take a raincoat

This is very common in Icelandic: the noun changes form depending on its grammatical role.

Because the preposition með normally takes the dative when it means with.

The pronoun forms of ég are:

• ég = I
• mig = me (accusative)
• mér = me (dative)
• mín = my / mine-related forms depending on use

So:

• með mér = with me

In this sentence, taka regnfrakka með mér means take a raincoat with me / bring a raincoat along.

Yes, it can be.

Literally, taka ... með mér means take ... with me, but in natural English that often corresponds to bring.

So:

• Ég hefði átt að taka regnfrakka með mér
  = literally I should have taken a raincoat with me = naturally I should have brought a raincoat

Icelandic often expresses this idea with taka ... með sér/mér/þér.

Í staðinn means instead or instead of that.

It is a fixed expression:

• í staðinn = instead
• en í staðinn = but instead

So the sentence contrasts two ideas:

• what the speaker should have done
• what the speaker actually did instead

It is not necessary here to add another noun after it; it works on its own as an adverbial phrase.

Because Icelandic is a verb-second language in main clauses.

That means the finite verb usually comes in the second position. When another element is placed first for emphasis or linking, the verb comes next, and the subject follows.

Here the clause begins with:

• í staðinn = instead

So the order becomes:

• Í staðinn fór ég ...
• literally: Instead went I ...
• natural English: Instead I went ...

This is very normal Icelandic word order.

Bara usually means just, only, or simply.

In this sentence it gives the sense of:

• I just went into the fitting room...
• I simply went into the fitting room...

It can make the action sound casual, a little dismissive, or like that was the simple thing the speaker ended up doing instead.

Mátunarklefann is accusative singular definite.

That happens because fara í takes:

• accusative when it means movement into
• dative when it means being in

So:

• fara í mátunarklefann = go into the fitting room
• vera í mátunarklefanum = be in the fitting room

This is an important Icelandic pattern with many prepositions: case can reflect movement versus location.

Also, the -ann ending is the suffixed definite article, meaning the.

Because annan jakka is the accusative masculine singular form, and it is the object of keypti.

The base form is:

• jakki = jacket

As a direct object, it becomes:

• jakka

And the adjective annar also has to match the noun in gender, number, and case:

• annar jakki = another jacket (nominative)
• annan jakka = another jacket (accusative)

So both words change together.

They are related but not identical words.

• regnfrakki = raincoat
• jakki = jacket

A regnfrakki is specifically for rain or wet weather. A jakki is more general. So the sentence is contrasting what the speaker should have brought with what they ended up buying.

Yes.

In Icelandic, once the subject is clear, it does not need to be repeated in the second part of a coordinated clause if the same subject continues.

So:

• fór ég ... og keypti ...

means:

• I went ... and bought ...

The subject ég applies to both verbs:

• fór
• keypti

Yes. That is often a very good way to understand it.

Depending on context:

• ég á að taka = I should take / I am supposed to take
• ég átti að taka = I was supposed to take
• ég hefði átt að taka = I should have taken / I was supposed to have taken

In this sentence, English should have taken is probably the most natural translation, because it sounds like regret about something not done.