Learn Hardcore Icelandic: Á laugardögum förum við stundum í verslunarmiðstöðina í miðbænum. - On Saturdays we sometimes go to the shopping mall downtown.

Á laugardögum = on Saturdays (in general) → dative plural
Á laugardag (more fully á laugardaginn) = on Saturday (one particular Saturday) → accusative singular

In Icelandic, when you say you usually / regularly do something on a certain day of the week, you normally use:

á mánudögum – on Mondays
á þriðjudögum – on Tuesdays
á laugardögum – on Saturdays

So Á laugardögum förum við… means On Saturdays we (usually) go…

What case is laugardögum, and what is the base word?

The base word is laugardagur (Saturday), a masculine noun.
In á laugardögum, laugardögum is dative plural.

A simplified declension of laugardagur:

Nom. sg: laugardagur – Saturday (as subject)
Acc. sg: laugardag / laugardaginn – on Saturday (specific)
Dat. sg: laugardegi
Gen. sg: laugardags
Nom. pl: laugardagar – Saturdays
Acc. pl: laugardaga
Dat. pl: laugardögum – on Saturdays (habitual)
Gen. pl: laugardaga

The preposition á takes the dative here because it is used for time expressions of when something regularly happens.

Could I also say Á laugardag förum við stundum í verslunarmiðstöðina…?

You could, but the meaning changes:

Á laugardögum förum við stundum…
→ On Saturdays (in general), we sometimes go… (habitual)
Á laugardag förum við stundum… is odd, because stundum (sometimes) suggests a general habit, but á laugardag (singular) points to one specific Saturday.

More natural uses:

Á laugardaginn förum við í verslunarmiðstöðina.
→ On Saturday we are going to the shopping mall. (this coming Saturday)

So for talking about a general routine, keep á + dative plural: á laugardögum.

Why is the word order Á laugardögum förum við… and not Á laugardögum við förum… or Við förum á laugardögum…?

Icelandic normally has a verb-second (V2) word order in main clauses: the finite verb (here förum) usually comes in second position, no matter what is in first position.

Here, Á laugardögum (a time phrase) is put first for emphasis.
Because something is already in first place, the verb must come second:
- Á laugardögum förum við stundum…

Both of these are correct and natural:

Á laugardögum förum við stundum í verslunarmiðstöðina… (time phrase first)
Við förum stundum á laugardögum í verslunarmiðstöðina… (subject first)

But *Á laugardögum við förum… is ungrammatical in standard Icelandic because it breaks the verb‑second rule.

What exactly does förum mean, and why isn’t it fara?

Fara is the infinitive form of the verb to go.
Förum is the 1st person plural present indicative:

ég fer – I go
þú ferð – you go (singular)
hann/hún fer – he/she goes
við förum – we go
þið farið – you go (plural)
þeir/þær/þau fara – they go

So förum goes specifically with við (we):
við förum = we go / we are going.

In this sentence the present tense expresses a habit:

Á laugardögum förum við… = On Saturdays we (normally) go…

What does stundum mean, and where can it go in the sentence?

Stundum means sometimes.

Common positions:

Á laugardögum förum við stundum í verslunarmiðstöðina…
Á laugardögum förum við í verslunarmiðstöðina stundum… (less usual)
Við förum stundum á laugardögum í verslunarmiðstöðina…

Typical, natural place is after the verb or after the subject:

Við stundum förum… sounds strange; you normally say
Við förum stundum…

So in your sentence, förum við stundum is perfectly normal word order:
verb (förum) – subject (við) – adverb (stundum).

Why is it í verslunarmiðstöðina and not just í verslunarmiðstöð?

Two main reasons:

Direction vs. location
The preposition í takes:
- accusative for movement into something,
- dative for location in something.
Here you are going into the mall, so you need accusative:
- í verslunarmiðstöðina – into the shopping mall (accusative definite)
- Compare: við erum í verslunarmiðstöðinni – we are in the mall (dative definite)
Definite article
verslunarmiðstöðina = verslunarmiðstöð
- -ina (definite ending) → the shopping mall.

So í verslunarmiðstöðina literally is into the shopping mall.

How is the word verslunarmiðstöðina built up, and what is its gender?

Verslunarmiðstöðina comes from the feminine noun verslunarmiðstöð (shopping mall, shopping centre).

It is a compound:

verslun – shop, trade
mið – middle
stöð – station/place

Together: something like “shopping‑middle‑place” → shopping centre.

Gender: feminine.

Relevant forms (singular):

Nom. indef.: verslunarmiðstöð – a shopping mall
Nom. def.: verslunarmiðstöðin – the shopping mall
Acc. indef.: verslunarmiðstöð
Acc. def.: verslunarmiðstöðina – the shopping mall (object, or motion into)

In the sentence, í verslunarmiðstöðina uses the accusative definite singular, because you are moving into the mall.

Why is it í miðbænum and not í miðbæinn or í miðbærinn?

The base noun is miðbær (city centre, downtown), masculine.

Location vs. direction
As with verslunarmiðstöð, the preposition í takes:
- dative for location in:
  í miðbænum – in the city centre → dative
- accusative for movement into:
  í miðbæinn – into the city centre → accusative
In your sentence, the mall already is in the city centre:
- …to the shopping mall in the city centre → a static location → dative miðbænum.
Definiteness and endings
Singular forms of miðbær:
- Nom. sg: miðbær
- Nom. sg definite: miðbærinn
- Acc. sg: miðbæ
- Acc. sg definite: miðbæinn
- Dat. sg: miðbæ
- Dat. sg definite: miðbænum

So í miðbænum is dative singular definite: in the city centre (of our town/city).

Could I say í miðbæ instead of í miðbænum? Does it change the meaning?

Yes, you can say í miðbæ, but there is a nuance:

í miðbænum – in the city centre (the specific, known one: usually your town’s centre)
í miðbæ – in a/the city centre more vaguely or generically, without stressing “the” specific one.

In everyday speech, if you are talking about your own town or city, í miðbænum is much more natural, because both speaker and listener know which city centre you mean.

Why is there í twice: í verslunarmiðstöðina í miðbænum? Could I drop one?

You need both prepositions because they introduce two different relationships:

í verslunarmiðstöðina – movement into the mall
í miðbænum – location in the city centre

What you are really saying is:

…into the shopping mall *that is located in the city centre.*

Each í belongs to its own phrase:

í verslunarmiðstöðina
í miðbænum

You cannot normally drop the second í.
If you tried í verslunarmiðstöðina miðbænum, it would be ungrammatical and unclear.

Could I leave out við and just say Á laugardögum förum stundum í verslunarmiðstöðina…?

No, not in standard Icelandic. You generally cannot drop subject pronouns the way you can in some languages (like Spanish or Italian).

You must say:

Á laugardögum förum við stundum í verslunarmiðstöðina í miðbænum.

If you omit við, it becomes ungrammatical or at least very unnatural. The only common exceptions are commands or imperative forms (where the subject “you” is understood), but this is a normal declarative sentence, so við is required.