Learn Hardcore Hungarian: Nekem is fontos ez az élmény, mert először utazom külföldi városba vonattal. - This experience is important for me too, because I am traveling by train to a foreign city for the first time.

Nekem is fontos puts stronger emphasis on “to me (too)”. It sounds like you’re contrasting yourself with others: “It’s important to me as well.”
Fontos nekem is a bit more neutral, with the stress more on fontos (“important”) itself; “It is important for me.”

In many contexts they’re interchangeable, but Nekem is fontos sounds more like you’re explicitly adding yourself to a group of people for whom this experience is important.

Why is it Nekem and not Én at the beginning? Why do we need the dative case here?

Hungarian normally uses the dative case (-nak/-nek) for the person for whom something is important:

Nekem fontos – It is important to me
Neked fontos – It is important to you
Neki fontos – It is important to him/her

Using Én fontos would be ungrammatical; én is nominative (“I”), and fontos here describes the experience, not the person. So you must mark the “experiencer” with the dative: nekem.

What exactly does is do in Nekem is fontos? Where does it have to go in the sentence?

is means “also / too / as well.” It always follows the word it is modifying:

Nekem is fontos – It is important *to me too.*
Ez az élmény is fontos – This experience *too is important (among others).*

If you move is, you move the focus of “too”:

Nekem is fontos ez az élmény – It’s important to me too (maybe it was already said that it’s important to others).
Nekem fontos ez az élmény is – Among several experiences, this one too is important to me.

So the position of is is crucial for what exactly is “also”.

Why do we say ez az élmény and not just ez élmény or az élmény?

With a demonstrative (ez / az) plus a noun, Hungarian normally uses both the demonstrative and the definite article:

ez a könyv – this book
az a ház – that house

Here, élmény starts with a vowel (é), so the form is:

ez az élmény – literally “this the experience” → just “this experience.”

You cannot say ez élmény; that sounds incomplete or wrong.
You could say only az élmény, but then you lose the pointing sense of “this”, and it just means “the experience” in general.

Why is it ez az élmény and not az az élmény, since “élmény” starts with a vowel?

The choice between ez and az is about distance / deictic “this vs that”, not about the first sound of the noun:

ez = this (near)
az = that (far)

The article that follows (a/az) does depend on the next word:

ez a ház (ház starts with consonant)
ez az élmény (élmény starts with vowel)

So:

ez az élmény = this experience
az az élmény = literally that that experience (“that experience over there / that specific one”).

In your sentence, ez az élmény is correct because the speaker means “this” experience, not “that”.

Could I say Ez az élmény nekem is fontos instead? Does changing the word order change the meaning?

Yes, Ez az élmény nekem is fontos is perfectly correct.

Word order in Hungarian mainly changes emphasis:

Nekem is fontos ez az élmény
– Emphasis on Nekem is: “To me too, this experience is important.” You’re adding yourself to others.
Ez az élmény nekem is fontos
– Emphasis more on Ez az élmény: “This experience is important to me as well (as other things / other experiences).”

The basic meaning is the same, but in natural speech you choose the order according to what you’re highlighting: the person (nekem) or the experience (ez az élmény).

Why is utazom in the present tense if the English translation is about the future?

Hungarian often uses the simple present to talk about the near future or arranged future actions, especially with adverbs like először, holnap, jövő héten:

Holnap utazom Budapestre. – I’m traveling to Budapest tomorrow.
Először utazom külföldi városba vonattal. – I’m traveling to a foreign city by train for the first time.

You can say fogok utazni (future with fog), but that tends to sound more explicit / heavy, and is often not needed in everyday speech. The present tense here already naturally implies a planned future trip.

I’ve also seen utazok. What’s the difference between utazom and utazok?

Both utazom and utazok are 1st person singular present of utazni (to travel), and both are understood as “I travel / I am traveling.”

utazom – considered more standard / formal (especially in writing).
utazok – common in colloquial speech, often used in everyday conversation.

In a sentence like this one, először utazom is stylistically a bit more careful/standard, but először utazok would also be heard in spoken Hungarian.

What does először do here exactly? Could I move it, for example: Külföldi városba először utazom vonattal?

először means “for the first time.”

először utazom – I’m traveling for the first time.

You can move it, but the focus and rhythm change:

Először utazom külföldi városba vonattal.
– Neutral, “For the first time I’m traveling to a foreign city by train.”
Külföldi városba először utazom vonattal.
– Emphasis shifts toward külföldi városba: “It’s to a foreign city that I’m traveling by train for the first time.” (Implying maybe you’ve already traveled by train domestically.)

All these orders can be grammatical, but először utazom is the most straightforward and natural in many contexts.

Why is it külföldi városba and not külföldi városban? What does the ending -ba mean?

The suffix -ba/-be indicates movement into something: “to / into.”

városba – to the city / into a city
városban – in the city (location, not movement)

So:

külföldi városba utazom – I’m traveling *to a foreign city.*
külföldi városban utazom would mean something odd like “I’m traveling *in a foreign city”* (moving around inside it), which is not the intended meaning here.

Therefore -ba is correct, because the verb utazom expresses movement towards a place.

Why külföldi városba and not just külföldre? What’s the difference?

Both are possible, but they have different nuances:

külföldre utazom – I’m traveling *abroad.* (General: to a foreign country/countries, not specifying what exactly.)
külföldi városba utazom – I’m traveling to a *foreign city.* (You specify that the destination is a city, and that city is in a foreign country.)

In your sentence, külföldi városba adds the extra detail that it’s a city abroad, not just “somewhere abroad”.

Why vonattal and not something like vonaton or a preposition like “by train”?

Hungarian does not use a preposition like English “by” here. Instead it uses the instrumental/comitative case -val/-vel, which often corresponds to “with / by (means of)”:

vonattal ← vonat + -val → by train / with a train
busszal – by bus
autóval – by car

vonaton (with -on/-en/-ön) would mean “on the train” (location), not “by train” as a means of transport.

So külföldi városba vonattal utazom literally is “I travel to a foreign city with train,” i.e. by train.

Why is it vonattal (with double t) and not vonatval?

The -val/-vel ending undergoes consonant assimilation:

If the noun ends in a consonant, the v of -val/-vel usually assimilates to that consonant and they are written doubled.

vonat + -val → vonattal
Steps:

The v of -val becomes similar to the preceding t,
The result is written with a double t: -ttal.

Other examples:

kéz + -vel → kézzel (with a hand)
híd + -val → híddal (with a bridge)

So vonattal is the correct, assimilated form.

There is no egy before külföldi városba. Shouldn’t it be egy külföldi városba for “to a foreign city”?

Hungarian often omits the indefinite article egy when the noun phrase is:

not emphasising “one (single)”, and
clearly indefinite from context.

So:

külföldi városba utazom – natural, I’m traveling to a foreign city.
egy külföldi városba utazom – also correct, but egy adds a nuance of “one particular foreign city” (or focuses that it is one, not several).

In neutral statements like this, it’s very common and natural not to use egy.

What is the nuance of élmény here? Is it just “experience” like any experience?

élmény is usually a personal, often emotionally significant experience, often something memorable or impactful (positive or negative):

Jó élmény volt. – It was a good (pleasant) experience.
Felejthetetlen élmény. – An unforgettable experience.

More neutral or technical “experience” can be tapasztalat (e.g. work experience, life experience in general):

Munkatapasztalat – work experience

In this sentence, ez az élmény suggests “this special / personal experience”, not just a routine or dry “experience” in the CV sense.