Ég ætla að kíkja í bakaríið eftir vinnu.

Ætla að + infinitive is a very common Icelandic pattern meaning to intend to, to be going to, or to plan to do something.

So:

• Ég ætla að kíkja = I’m going to stop by / I intend to pop in

Even though ætla is in the present tense, the whole phrase often refers to a future action, just like English I’m going to....

Here að is the infinitive marker, like English to before a verb.

• kíkja = to peek / to look / to stop by
• að kíkja = to stop by

After ætla, Icelandic normally uses að + infinitive:

• Ég ætla að fara = I’m going to go
• Ég ætla að borða = I’m going to eat
• Ég ætla að kíkja = I’m going to stop by

So this að is not the same as the preposition að meaning to/toward.

Yes, kíkja can literally mean peek or look, but in everyday speech it often has a broader meaning like:

• stop by
• pop in
• have a quick look
• drop in briefly

In this sentence, kíkja í bakaríið most naturally means something like:

• stop by the bakery
• pop into the bakery

So it often suggests a quick, casual visit rather than a long stay.

The combination kíkja í is very common and often means look into, pop into, or check out something.

Examples:

• kíkja í búð = pop into a shop
• kíkja í bók = look in a book
• kíkja í bakaríið = pop into the bakery

The preposition í often gives the sense of into or inside. With places, it can suggest entering briefly.

Bakarí means a bakery.
Bakaríið means the bakery.

Icelandic usually adds the definite article as a suffix to the noun rather than using a separate word like English the.

So:

• bakarí = a bakery
• bakaríið = the bakery

That -ið is the neuter singular definite ending here.

This is about case and the difference between motion and location after í.

With í:

• accusative is usually used for motion into
• dative is usually used for being in / location

So:

• Ég fer í bakaríið = I go into the bakery
• Ég er í bakaríinu = I am in the bakery

In your sentence, the idea is going into / stopping by the bakery, so accusative is used: í bakaríið.

Because eftir governs the dative case in this meaning.

The noun is:

• vinna = work

Its dative singular form is:

• vinnu

So:

• eftir vinnu = after work

This is a very useful everyday phrase.

In eftir vinnu, vinna usually means work in the general sense, as in:

• after work
• when I finish work

It does not necessarily mean the physical workplace itself. It is similar to English after work rather than after the workplace.

Grammatically, ætla is in the present tense:

• ég ætla = I intend / I am going to

But the meaning is usually future-oriented. So the sentence refers to something the speaker plans to do later:

• I’m going to stop by the bakery after work

This is very normal in Icelandic. Present-tense forms are often used to talk about future plans.

Yes. Icelandic word order is flexible, though not completely free.

Your sentence:

• Ég ætla að kíkja í bakaríið eftir vinnu.

A very natural variation is:

• Eftir vinnu ætla ég að kíkja í bakaríið.
  = After work, I’m going to stop by the bakery.

Notice that when Eftir vinnu comes first, the finite verb ætla comes before ég. That is a standard Icelandic word-order pattern.

Yes, very natural. It sounds like normal conversational Icelandic.

It uses several common everyday patterns:

• ég ætla að... = I’m going to...
• kíkja í... = pop into...
• eftir vinnu = after work

So this is exactly the kind of sentence a native speaker might say casually in daily life.

Then you would need a verb of location, not motion, and you would use dative after í.

For example:

• Ég er í bakaríinu eftir vinnu.
  = I am in the bakery after work.

Compare:

• Ég ætla að kíkja í bakaríið eftir vinnu.
  = I’m going to pop into the bakery after work.
  -> motion into the bakery

• Ég er í bakaríinu eftir vinnu.
  = I am in the bakery after work.
  -> location inside the bakery

This motion/location contrast is very important in Icelandic.