Á mánudögum fer gönguhópurinn í kvöldgöngu í skóginum.

Icelandic often uses the dative plural of days of the week to mean “on Mondays / every Monday” (a repeated, habitual action).

• The base form is mánudagur (Monday, nominative singular).
• The dative plural is mánudögum.

With the preposition á and a day of the week:

• á mánudögum = on Mondays (regularly, every Monday)
• á mánudegi = on Monday (one specific Monday)

So Á mánudögum tells you this is a recurring activity.

In this sentence, Á means “on” in a time expression.

• With time phrases like days of the week, á usually takes the dative case.
• That’s why we get á mánudögum (dative plural), not á mánudagar or á mánudaga.

So Á mánudögum literally is “on Mondays (in the dative case)”.

Icelandic has a “verb-second” (V2) word order in main clauses, similar to German:

1. Some element comes first (here, the time phrase Á mánudögum).
2. The finite verb comes second (fer).
3. The subject then follows (gönguhópurinn).

So:

• Á mánudögum fer gönguhópurinn… = On Mondays goes the walking group…
• If nothing is fronted, you can also say Gönguhópurinn fer á mánudögum…, which is also correct.

Both are correct; the version in the sentence just fronts the time expression for emphasis / flow.

Fer is the 3rd person singular present of fara (“to go, to leave, to travel”).

• fara is a general movement verb: to go, to head off somewhere.
• ganga is more specifically “to walk” (as a verb).

In Icelandic, it’s very common to say fara í + [activity], for example:

• fara í sund – to go swimming
• fara í bíó – to go to the cinema
• fara í göngu / kvöldgöngu – to go for a (evening) walk

So the sentence uses the very natural pattern fer í kvöldgöngu = “goes for an evening walk”, not gengur í kvöldgöngu.

Gönguhópurinn is a compound noun + definite article:

• ganga = walk
• göngu- = genitive form used in compounds (roughly “of walking”)
• hópur = group
• gönguhópur = a walking group / hiking group
• -inn = “the” (definite article, masculine singular, attached to the noun)

So gönguhópurinn literally is “the walking group”.
In Icelandic, the definite article is usually a suffix (attached to the end of the noun) rather than a separate word like the in English.

The base noun is kvöldganga (“evening walk”).

• In the singular, its accusative form is kvöldgöngu.
• The change a → ö is a regular sound change (u‑umlaut) in some forms.

The preposition í can take accusative (movement, going into/to something) or dative (location, being in something).

Here, í kvöldgöngu is í + accusative, following the very common pattern fara í + [activity] (“go to/for [activity]”):

• í kvöldgöngu = for an evening walk (activity, goal), accusative.

Yes, both í are doing different jobs:

• í kvöldgöngu (with accusative) = for an evening walk (the activity / goal)
• í skóginum (with dative) = in the forest (the place where it happens)

If you removed one í, you would either:

• Lose the idea of “going for a walk” (if you dropped the first í), or
• Lose the idea that this happens in the forest (if you dropped the second í).

So the pattern is:
fer [í kvöldgöngu] [í skóginum]
= goes [for an evening walk] [in the forest].

The base noun is skógur (forest, nominative singular, masculine).

• The dative singular is skógi.
• With the definite article ( “the” ) it becomes skóginum (dative singular definite).

The preposition í takes dative for location / being in a place:

• í skógi – in a forest
• í skóginum – in the forest

So í skóginum is “in the forest”, using dative + definite article to indicate a specific forest.

Yes, that sentence is also grammatical.

Core rule: the finite verb stays in second position; other elements (time, place, object) can be rearranged for emphasis or style:

• Á mánudögum fer gönguhópurinn í kvöldgöngu í skóginum.
  – Emphasis starts with when (On Mondays…).
• Gönguhópurinn fer í kvöldgöngu í skóginum á mánudögum.
  – Emphasis starts with who (The walking group…).

Both mean essentially the same thing; word order mainly affects information focus.

Icelandic normally uses the simple present where English often uses the present progressive.

So:

• Gönguhópurinn fer… can mean
  – “The walking group goes …” (regularly), or
  – “The walking group is going …” (right now), depending on context.

The language does not form the progressive with “to be” + present participle the way English does.
Aspect (habitual vs right-now) is usually understood from context, adverbs, and time expressions like Á mánudögum.