Learn Hardcore Icelandic: Gítar og píanó eru tvö hljóðfæri sem hún lærir á. - Guitar and piano are two musical instruments that she is learning to play.

Even though gítar and píanó each look like singular nouns, together they form a plural subject:
Gítar og píanó = guitar and piano → “they” (two instruments)

So the verb must agree in number:
Gítar og píanó eru … = Guitar and piano are …

Why is it “tvö” and not “tveir” or “tvær”?

Icelandic has three genders and the number “two” changes form to match the gender of the noun:

tveir – masculine
tvær – feminine
tvö – neuter

The word hljóðfæri (instrument) is neuter, so “two instruments” must be:

tvö hljóðfæri

That’s why the sentence has “tvö” and not tveir or tvær.

Why does “hljóðfæri” look the same in the singular and the plural?

Hljóðfæri is a neuter noun of a type that has the same form in the singular and plural in the nominative and accusative:

Singular nominative: hljóðfæri – an instrument
Plural nominative: hljóðfæri – instruments

You see the number from the context, for example:

Eitt hljóðfæri – one instrument
Tvö hljóðfæri – two instruments

With the definite article, the forms differ slightly:

Singular: hljóðfærið – the instrument
Plural: hljóðfærin – the instruments

What is the role of “sem” in this sentence?

Sem is a relative pronoun (like English “that/which/who”).

The sentence is structured as:

Gítar og píanó eru tvö hljóðfæri – Guitar and piano are two instruments
sem hún lærir á – which she is learning (to play)

So sem refers back to hljóðfæri (instruments) and introduces the relative clause:

… tvö hljóðfæri sem hún lærir á.
… two instruments which she is learning (to play).

Why is the preposition “á” at the end: “sem hún lærir á” instead of before hljóðfæri?

This is preposition stranding, similar to English:

English: instruments which she is learning to play on → preposition on at the end
Icelandic: hljóðfæri sem hún lærir á → preposition á at the end

Literally:

… instruments which she learns on.

When you have a relative clause (sem hún lærir á), the preposition often ends up stranded at the end of the clause, exactly as in many informal English sentences (“the instrument she plays on”).

Why do we need “á” with “lærir”? Can we say just “sem hún lærir”?

With musical instruments, Icelandic normally uses “læra á + [hljóðfæri]” to mean “learn to play [an instrument]”.

að læra á gítar – to learn (to play) the guitar
að læra á píanó – to learn (to play) the piano

Literally it’s “learn on [instrument]”.

Without á, læra just means “learn” in a more general sense (learn math, learn a text, etc.). In this sentence, to make it clear she is learning to play the instruments, you need á:

… sem hún lærir á.
… which she is learning (to play).

Why doesn’t the sentence repeat the nouns after “á”, like “sem hún lærir á þau”?

It’s not necessary to repeat them, because “sem” already refers back to hljóðfæri:

hljóðfæri sem hún lærir á
literally: instruments which she learns on (them)

Saying “á þau” (on them) would be grammatically possible but sounds unnecessary and heavier in this context. Icelandic, like English, often omits pronouns that are clearly understood.

Compare in English:

instruments which she is learning (to play)
rather than
instruments which she is learning to play them (incorrect in standard English).

What case are “gítar” and “píanó” in here?

They are in the nominative case, because they form the subject of the sentence:

Gítar og píanó eru tvö hljóðfæri …

Some details:

gítar – a regular masculine noun, nominative singular form here
píanó – an indeclinable neuter loanword; it looks the same in several cases

Although each is formally singular, the phrase “gítar og píanó” together counts as plural, so the verb is eru and the complement is plural: tvö hljóðfæri.

Why is there no word for “the” before gítar and píanó?

Icelandic normally marks definiteness with a suffix rather than a separate word:

gítar – a guitar
gítarinn – the guitar
píanó – a piano
píanóið – the piano

In this sentence, the instruments are mentioned in a general / indefinite way, more like:

Guitar and piano are two instruments she is learning.

If you wanted “the guitar and the piano”, you could say:

Gítarinn og píanóið eru tvö hljóðfæri sem hún lærir á.

But that changes the meaning to those specific instruments.

What is the function of “hún” and why not “henni”?

hún = “she” (nominative, subject form)
henni = “her” (dative/indirect object form)

In “sem hún lærir á”, hún is the subject of the verb lærir:

(hljóðfæri) sem hún lærir á
(instruments) which she is learning (to play)

So it must be in the nominative: hún, not henni.

Is “lærir” present tense? What is its full verb form?

Yes, lærir is the 3rd person singular present tense of the verb að læra (to learn).

Basic forms of að læra:

Infinitive: að læra – to learn
1st sg. pres.: ég læri – I learn
2nd sg. pres.: þú lærir – you learn
3rd sg. pres.: hann/hún/það lærir – he/she/it learns

So hún lærir = she learns / she is learning.

How would the sentence change if the learner were male instead of female?

You only need to change the pronoun hún (she) to hann (he). Everything else stays the same:

Gítar og píanó eru tvö hljóðfæri sem hann lærir á.
Guitar and piano are two instruments which he is learning (to play).

How are the main words in this sentence pronounced (roughly)?

Approximate pronunciations (using English-like hints):

Gítar – [GEE-tar], with a long í like see
og – often [oh] in speech (the g is usually not fully pronounced)
píanó – [PEE-a-no], stress on the first syllable
eru – [EH-ru], short e as in get
tvö – [tvuh], rounded vowel (like u in French tu; no exact English equivalent)
hljóðfæri – roughly hlj sounds like hly together
- ð is a soft th as in this
sem – [sem], like sem in semolina
hún – [hoon], long oo as in moon
lærir – [LAI-rir], æ like eye
á – [ow], as in English how

These are only rough guides, but they’re close enough for a learner to be understood.