Reggel zuhanyozom, utána egy csésze kávét iszom.

Here reggel functions as an adverb meaning in the morning.

Hungarian often uses bare time words this way:

• reggel = in the morning
• este = in the evening
• éjjel = at night

So you do not need a word like in here, and you do not need an article.

A few useful comparisons:

• Reggel zuhanyozom. = I shower in the morning.
• Ma reggel zuhanyoztam. = I showered this morning.
• Reggelente zuhanyozom. = I shower in the mornings / every morning.

Hungarian usually leaves out subject pronouns when the verb ending already shows who the subject is.

So:

• zuhanyozom already means I shower
• iszom already means I drink

Because of that, én is not necessary.

You can add én if you want emphasis or contrast:

• Én reggel zuhanyozom. = I shower in the morning.
• Én iszom a kávét, nem ő. = I am drinking the coffee, not him/her.

The dictionary forms are:

• zuhanyozik = to shower
• iszik = to drink

These are -ik verbs, so their I form often looks different from the dictionary form:

• zuhanyozik → zuhanyozom
• iszik → iszom

So in this sentence:

• zuhanyozom = I shower
• iszom = I drink

This is very common with Hungarian verbs: the form you see in a sentence is often not the same as the dictionary form.

Because it is the direct object of iszom.

Hungarian marks direct objects with the accusative ending -t:

• kávé = coffee
• kávét = coffee, as the thing being drunk

So:

• Kávét iszom. = I drink coffee.

The same idea appears in many nouns:

• tea → teát
• víz → vizet
• alma → almát

In Hungarian, with measure/container expressions, the accusative usually appears on the thing being measured, not on the container word.

So:

• egy csésze kávét = a cup of coffee
• egy pohár vizet = a glass of water
• két kiló almát = two kilos of apples

That is why kávét gets the -t, while csésze stays unchanged.

If you were talking about the physical cup itself as the direct object, then csészét could appear:

• Eltörtem egy csészét. = I broke a cup.

But in egy csésze kávét iszom, the meaning is I drink a cupful of coffee, so kávét is the marked word.

Here utána means after that or afterwards.

It links the two actions in sequence:

1. Reggel zuhanyozom
2. utána egy csésze kávét iszom

So the sense is:

• first I shower
• then/after that I drink a cup of coffee

It is related to után, which means after:

• ebéd után = after lunch
• utána = after it / after that / afterwards

In everyday sentences, utána is very commonly used just as afterwards.

Yes, Hungarian word order is flexible, but it is not random. It often reflects topic, focus, and what is new information.

In this sentence:

• utána sets the time/order frame: after that
• egy csésze kávét is the object
• iszom is the verb

So Utána egy csésze kávét iszom is a natural, neutral sentence.

Other orders are possible, but the emphasis changes. For example:

• Utána kávét iszom. = Afterwards I drink coffee.
• Egy csésze kávét iszom utána. = It is a cup of coffee that I drink afterwards.

For a beginner, the safest takeaway is:

• Hungarian often allows several word orders
• changing the order often changes the emphasis, not the basic meaning

It can be both.

Hungarian uses egy for:

• the number one
• the indefinite article a/an

So egy csésze kávé can mean:

• a cup of coffee
• or more emphatically one cup of coffee

In this sentence, the most natural reading is simply a cup of coffee, though it can also lightly suggest one cup.

Compare:

• iszom kávét = I drink coffee
• egy csésze kávét iszom = I drink a cup of coffee

Yes, you could use reggelente, but the nuance changes slightly.

• reggel = in the morning
• reggelente = in the mornings / mornings generally / every morning-ish

So:

• Reggel zuhanyozom. = I shower in the morning.
• Reggelente zuhanyozom. = I shower in the mornings.

The version with reggelente sounds more clearly habitual or repeated. The version with reggel can also be habitual, but it is a bit more general and context-dependent.

A few important ones:

• cs in csésze sounds like ch in chair
• sz in csésze sounds like s in see
• zs in zuhanyozom sounds like the s in measure
• ny in zuhanyozom is like ny in canyon for many English speakers
• gy in egy is a soft sound somewhat like dy in during for many speakers

A very rough pronunciation guide:

• Reggel ≈ REG-gel
• zuhanyozom ≈ ZOO-ha-nyo-zom
• utána ≈ oo-TAA-na
• egy ≈ a short soft edy/dy sound
• csésze ≈ CHAY-se
• kávét ≈ KAA-vayt
• iszom ≈ EE-som

Also, Hungarian stress is usually on the first syllable of the word.

Yes. They tell you several things at once:

• person: first person
• number: singular
• tense: present

So:

• zuhanyozom = I am showering / I shower
• iszom = I am drinking / I drink

Hungarian present tense can often cover both the simple present and present continuous, depending on context.

That means this sentence can describe:

• a routine
• what someone is doing now
• a general habit

Context decides which reading is most natural.