Learn Hardcore Icelandic: Ég held áfram að læra íslensku á kvöldin. - I continue to learn Icelandic in the evenings.

Supine (used with hef etc.): haldið
- Ég hef haldið áfram að læra. = I have continued to learn.
Past participle (passive): also haldið.

Why is að used before læra? Is it like English “to”?

Yes. Here að is the infinitive marker, very close in function to English “to”:

að læra = to learn

After halda áfram (“continue”), you normally follow with að + infinitive:

Ég held áfram að læra. = I continue to study.
Hann heldur áfram að vinna. = He keeps on working.

This að is not the same as að meaning “that” in clauses like:

Ég held að hann komi. = I think that he will come.

So in your sentence:

að læra = to learn (infinitive phrase, object of “continue”).

Why is it íslensku here and not íslenska?

Because íslenska is in the nominative, but in this sentence it is a direct object, so it must be in the accusative.

íslenska = Icelandic (the language), nominative.
After að læra (“to learn”), the thing you learn is the object, so you use accusative.
For this type of feminine noun, the accusative singular ends in -u, so íslenska → íslensku.

Compare:

Íslenska er erfitt mál.
“Icelandic is a difficult language.” (subject → nominative íslenska)
Ég læri íslensku.
“I learn/am learning Icelandic.” (object → accusative íslensku)

In your sentence, íslensku is the object of að læra, so accusative is required.

Which case is íslensku in exactly, and what does the paradigm of íslenska look like?

Here íslensku is accusative singular, because it’s the direct object of að læra.

Íslenska is a weak feminine noun (like stúlka “girl”). Its singular forms are:

Nominative: íslenska
– Íslenska er skemmtilegt tungumál.
“Icelandic is a fun language.”
Accusative: íslensku
– Ég læri íslensku.
“I learn Icelandic.”
Dative: íslensku
– Ég tala á íslensku.
“I speak in Icelandic.”
Genitive: íslensku
– Kennslubók í íslensku.
“A textbook in Icelandic.”

Accusative, dative and genitive singular all look the same (íslensku), so you determine the case by function and preposition, not by the ending alone. In your sentence, it’s accusative because of the verb læra (“to learn”).

What does á kvöldin literally mean, and why is kvöldin plural and definite?

Literally, á kvöldin is:

á = on / at (here: “in (the evenings)” as a time expression)
kvöldin = “the evenings” (definite plural of kvöld, “evening”)

So literally: “on the evenings”, but idiomatically “in the evenings”.

In Icelandic, a very common way to talk about a habitual time of day is:

á + [definite plural noun of time]

Examples:

á morgnana = in the mornings
á daginn = in the daytime / during the day
á kvöldin = in the evenings

The definite plural here (kvöldin, “the evenings”) expresses a repeated, general time frame, not specific individual evenings. This is why you don’t just say *á kvöld in this meaning; the idiomatic form is á kvöldin.

What’s the difference between á kvöldin and í kvöld (and maybe um kvöldin)?

They all refer to evening, but with different nuances:

á kvöldin
= “in the evenings”, habitual / repeated.
Used for things you usually do at that time:
- Ég held áfram að læra íslensku á kvöldin.
  I keep learning Icelandic in the evenings (regular habit).
í kvöld
= “this evening / tonight”, one specific evening (usually today):
- Ég ætla að læra íslensku í kvöld.
  I’m going to study Icelandic tonight.
um kvöldin
also often means “in the evenings”, similar to á kvöldin, with a slightly more “around/during the evenings” feel:
- Fólk horfir oft á sjónvarp um kvöldin.
  People often watch TV in the evenings.

For your sentence, because it describes an ongoing habit, á kvöldin is the natural choice.

What case does kvöldin take after á, and does á always take that case?

The preposition á can govern either accusative or dative, depending on meaning (movement vs location/time, and some fixed phrases).

Here, á kvöldin uses the accusative plural definite of kvöld:

kvöld = evening (nom/acc sg or pl, indefinite)
kvöldin = the evenings (definite nom/acc plural)

So in á kvöldin, á is taking the accusative.

Very roughly:

á + accusative: often movement onto/into something, or certain time expressions:
- Hann fór á fjallið. = He went onto the mountain.
- á daginn, á kvöldin = in the day, in the evenings (habitual time).
á + dative: often static location:
- Hann er á fjallinu. = He is on the mountain.
- Ég er á Íslandi. = I am in Iceland.

With time expressions like á kvöldin, you mainly just learn the idiom as a chunk: á + [time word in the appropriate form].

Can the word order be changed (for example, putting á kvöldin at the beginning), and what happens to áfram?

Yes, Icelandic word order is somewhat flexible, but the finite verb must stay in second position (V2 rule), and halda and áfram usually stay close together.

Your original:

Ég held áfram að læra íslensku á kvöldin.

Possible variants:

Á kvöldin held ég áfram að læra íslensku.
- Time phrase á kvöldin is fronted for emphasis (“In the evenings, I continue to learn Icelandic.”).
- Verb held is still in second position.
- This is very natural.
Ég held áfram á kvöldin að læra íslensku.
- Grammatically possible, but sounds heavier/less natural.
- Emphasis shifts slightly toward continuing in the evenings rather than just when you learn.

What you normally don’t do:

*Ég held að læra áfram íslensku á kvöldin.
This is wrong for your meaning. Here held að would be read as “think that”, and að læra would start a different clause.

So:

Keep held in second position.
Keep áfram closely attached to halda: held áfram.
You can move á kvöldin (and other adverbials) around, especially to the front.

How would you say this sentence in the past tense?

You only need to change the form of halda from present held to past hélt:

Ég hélt áfram að læra íslensku á kvöldin.
= “I continued / I kept on learning Icelandic in the evenings.”

Everything else (the infinitive að læra, íslensku, á kvöldin) stays the same.

How do you negate this sentence naturally?

The usual negation word is ekki, which goes after the finite verb:

Ég held ekki áfram að læra íslensku á kvöldin.
= “I do not continue to learn Icelandic in the evenings.”

Here you’re negating the continuation itself.

If you instead say:

Ég held áfram að læra ekki íslensku á kvöldin.

it sounds like “I continue to learn not Icelandic in the evenings” (i.e. I’m still learning something, just specifically not Icelandic), which is a very different and rather odd meaning.

If what you mean is simply “I stopped learning Icelandic in the evenings,” a more natural positive verb would be:

Ég hætti að læra íslensku á kvöldin.
= “I stopped learning Icelandic in the evenings.”