Learn Hardcore Icelandic: Fyrsti dagurinn var góður.

fyrsti – the ordinal adjective first, here in masculine nominative singular weak form, agreeing with dagurinn.
dagurinn – the day, masculine noun dagur in nominative singular definite form (stem dagur
- suffixed article -inn).
var – past tense of the verb vera (to be), 3rd person singular (was).
góður – the adjective good, masculine nominative singular strong form, used as a predicative complement (after to be).

Why is it dagurinn and not just dagur for day?

Icelandic usually expresses the by attaching a suffixed article to the noun:

Indefinite: dagur = a day / day
Definite: dagurinn = the day

So dagurinn is dagur (nom. sg.) + the definite article -inn.
In this sentence, you are talking about a specific, known day (the first one), so Icelandic uses the definite form dagurinn.

What case is dagurinn in, and why?

Dagurinn is in the nominative singular:

It is the subject of the sentence.
With the verb vera (to be), the subject and the complement (the “thing after to be”) both appear in the nominative in Icelandic.

So:

Fyrsti dagurinn – subject in nominative
var góður – was good (with góður also in nominative).

Why is it fyrsti and not fyrstur or fyrsta?

The form of the adjective/ordinal must agree with the noun in:

Gender: dagur is masculine.
Number: singular.
Case: nominative.
Definiteness: the noun is definite (dagurinn).

Icelandic adjectives (including ordinal numbers) have strong and weak declensions:

Strong masculine nominative sg. of first: fyrstur
Weak masculine nominative sg. of first: fyrsti

When the noun is definite (dagurinn), the preceding adjective normally takes the weak ending.
Therefore:

fyrsti dagurinn = the first day (correct)
fyrstur dagurinn would be ungrammatical in normal usage.

Why is it góður and not góði, góðan, or góðurinn?

Góður is acting as a predicative adjective (after the verb vera, describing the subject), not as an adjective directly before a noun. For predicative adjectives:

They agree in gender, number, case with the subject.
They are normally in the strong form.
They do not take the definite article.

Here:

Subject: dagurinn – masculine, singular, nominative.
Predicative adjective: góður – strong masculine nominative singular.

Some contrasts:

góður dagur – a good day (attributive, strong)
góði dagurinn – the good day (attributive, weak + definite noun)
Dagurinn var góður – The day was good (predicative, strong, no article on the adjective).

What tense is var, and what is the present-tense form?

Var is the past tense (simple past) of vera (to be), 3rd person singular:

hann / dagurinn var – he / the day was

Present tense forms of vera include:

ég er – I am
þú ert – you are (sg.)
hann / hún / það er – he / she / it is
við erum – we are
þið eruð – you are (pl.)
þeir / þær / þau eru – they are

So:

Fyrsti dagurinn er góður – The first day is good (present)
Fyrsti dagurinn var góður – The first day was good (past).

Could I say Fyrsti dagur var góður instead? What would be the difference?

You can say Fyrsti dagur var góður, but it sounds less natural in most contexts.

Fyrsti dagurinn var góður – sounds like you are talking about a particular, known first day (for example, your first day at work, your first day in Iceland, etc.). This is the normal way to say it.
Fyrsti dagur var góður – grammatically possible, but feels more like “a first day was good” or a more generic description; in most real situations, speakers prefer the definite form.

In practice, when you have a specific “first day” in mind, you almost always use dagurinn with the article.

Can I change the word order, for example Dagurinn fyrsti var góður?

Not in ordinary, neutral Icelandic. The normal order is:

[Ordinal/Adjective] + [Noun]
→ Fyrsti dagurinn

Putting fyrsti after the noun (Dagurinn fyrsti) is not standard; it might only appear in very marked, poetic, or archaic language.

For the whole sentence, the neutral word order is:

Fyrsti dagurinn var góður.

Changing it to things like Góður var fyrsti dagurinn is possible, but now you are doing stylistic emphasis (fronting góður). For a learner, stick to Fyrsti dagurinn var góður.

How would I say My first day was very good in Icelandic based on this sentence?

You can build on the same structure:

Fyrsti dagurinn minn var mjög góður.

Breakdown:

fyrsti – first (weak masc. nom. sg., agreeing with dagurinn)
dagurinn – the day (definite nominative)
minn – my (masculine nominative singular form)
var – was
mjög – very
góður – good (strong masc. nom. sg.)

Word order with possessives is usually [noun + article] + [possessive pronoun]:
dagurinn minn = my day.

How is fyrsti dagurinn var góður pronounced, especially ð and -inn?

Approximate pronunciation (main stress on the first syllable of each content word):

Fyrsti – roughly FIR-sti
- y like the vowel in bit but a bit tenser.
- r is rolled or tapped.
- st is as in English st.
dagurinn – roughly DAH-gu-rin
- Initial d is often pronounced like a t in this position.
- g between vowels is a soft “gh” sound, like Spanish g in lago.
- The -inn is a clear -in sound; it is the definite article.
var – roughly vahr (short vowel).
góður – roughly GOH-thur
- ó is a long o like in go.
- ð is like the th in this (voiced).
- Final ur is unstressed and quite short.

So the whole sentence would sound roughly like:
FIR-sti DAH-ghu-rin var GOH-thur.