Eftir ferðina fórum við í íþróttahúsið að horfa á fótbolta.

Because Icelandic usually follows a verb-second pattern in main clauses.

That means:

• one element comes first,
• then the finite verb comes second.

Here, Eftir ferðina is placed first for emphasis or to set the time frame, so the verb fórum has to come next, and the subject við comes after it.

So:

• Við fórum ... = neutral starting point
• Eftir ferðina fórum við ... = After the trip, we went ...

Both can be possible, but the sentence you have is a very normal Icelandic structure.

Because ferðina is:

• the noun ferð = trip/journey
• in the accusative singular
• with the definite article attached, so it means the trip

The preposition eftir here takes the accusative, so you need ferðina, not nominative ferðin.

A quick comparison:

• ferðin = the trip as a subject
• ferðina = the trip after a preposition or as an object in accusative contexts

It is the Icelandic way of saying the.

Icelandic usually does not use a separate word like English the. Instead, the definite article is attached to the end of the noun.

So:

• ferð = a trip / trip
• ferðin = the trip
• ferðina = the trip in accusative singular

The exact ending changes depending on:

• gender
• number
• case

Fórum is the 1st person plural past tense of fara, which means to go.

So:

• ég fór = I went
• við fórum = we went

The infinitive is fara.

This is an irregular verb, so the past form does not look very similar to the infinitive.

Yes, it is completely correct.

The first í is the preposition into / to / in.

The second í is just the first letter of the noun íþróttahúsið.

So this is simply:

• í = into/to
• íþróttahúsið = the sports hall

It may look strange at first, but it is not a typo.

Because Icelandic uses different cases after í depending on whether there is:

• movement to/into something → accusative
• location in something → dative

Here, the sentence describes going to the sports hall, so it uses the accusative:

• í íþróttahúsið = into/to the sports hall

If the meaning were in the sports hall as a location, you would expect:

• í íþróttahúsinu = in the sports hall

This is a very important Icelandic pattern.

It is a compound noun:

• íþrótt = sport
• hús = house/building
• íþróttahús = sports hall / gym building
• íþróttahúsið = the sports hall

Icelandic makes lots of compound nouns, so learning to spot the parts is very useful.

Here að is the infinitive marker, similar to English to in to watch.

So:

• horfa = watch / look
• að horfa = to watch

In this sentence, it expresses purpose:

• they went to the sports hall to watch football

So að horfa tells you what they went there in order to do.

Yes, fara að + infinitive can often mean begin to.

For example:

• Hann fór að hlæja = He began to laugh

But in your sentence, that is not the best interpretation.

Why not? Because the structure is:

• fórum ... í íþróttahúsið að horfa ...

The destination í íþróttahúsið makes it clear that this is a movement + purpose sentence:

• we went to the sports hall to watch football

So here að horfa means purpose, not began watching.

Because the Icelandic verb horfa normally uses the preposition á when it means watch / look at.

So:

• horfa á sjónvarp = watch TV
• horfa á mynd = watch a film
• horfa á fótbolta = watch football

This is something you should learn as a unit:

• horfa á = watch / look at

Many Icelandic verbs regularly go with a particular preposition.

Because fótbolta is the accusative singular form of fótbolti.

That happens because horfa á takes its object in the accusative.

So:

• fótbolti = football, as a dictionary form
• fótbolta = football, as the object after á in this phrase

This is a common pattern with Icelandic nouns: the form changes depending on grammatical case.

Because Icelandic, like English, often uses a bare singular noun for a general activity or type of thing.

So horfa á fótbolta means:

• watch football
• watch soccer
• watch a football match / football in general

It does not have to mean one specific football.

If you wanted something more specific, Icelandic could make that clearer in other ways, for example with:

• a definite form
• a plural form
• another noun such as leik for match/game

Here the simple bare singular is natural and idiomatic.