Learn Hardcore Hungarian: Este szeretek megnézni egy érdekes filmet. - In the evening I like to watch an interesting movie.

It can mean this evening / tonight (referring to today’s evening), especially in everyday speech.
It can also mean in the evening / in the evenings in a more habitual, general sense, which is how it fits your sentence:
Este szeretek megnézni egy érdekes filmet.
→ In the evening I like to watch an interesting movie / In the evenings I like to watch…

If you want to be clearly habitual, you can also say:

Esténként szeretek megnézni egy érdekes filmet.
(In the evenings I like to watch… – explicitly “on evenings, as a habit”)

If you want to emphasize this particular evening, you’d say:

Ma este szeretek megnézni egy érdekes filmet.
(This evening I’d like to watch an interesting movie.)

Why is there no article before este? Why not az este?

When parts of the day (like reggel, délután, este) are used as adverbials of time (answering “when?”), Hungarian usually drops the article:

Reggel kávét iszom. – I drink coffee in the morning.
Délután tanulok. – In the afternoon I study.
Este szeretek megnézni egy érdekes filmet. – In the evening I like to watch an interesting movie.

You would use an article if you talk about “the evening” as a noun/thing, not as “when”:

Az este hosszú volt. – The evening was long.
Az estéről beszélünk. – We are talking about the evening.

Can the word order change? Is Szeretek este megnézni egy érdekes filmet also correct?

Yes, it is correct, and Hungarian allows several natural variants:

Este szeretek megnézni egy érdekes filmet.
Szeretek este megnézni egy érdekes filmet.
Egy érdekes filmet szeretek este megnézni.

All are grammatical. The neutral choice here is usually something like:

Este szeretek megnézni egy érdekes filmet.
or
Este szeretek egy érdekes filmet megnézni.

Main point: Hungarian word order is flexible, but it changes focus (what is emphasized). Putting este at the beginning makes “in the evening” the setting/background for the whole statement. Putting egy érdekes filmet at the front would emphasize what you like to watch.

Why do we say szeretek megnézni? Why not just szeretek nézni?

Both can exist, but they have slightly different nuances:

Szeretek nézni egy érdekes filmet.
Literally I like watching an interesting movie (focus on the activity of watching, more open‑ended).
Szeretek megnézni egy érdekes filmet.
Literally I like to watch an interesting movie (and finish it) (focus on watching it through to completion).

The verb nézni = to watch / to look at, ongoing action.
Megnézni = to watch / see something through, to watch (it) all the way, thanks to the prefix meg-, which adds a sense of completeness.

In your sentence, megnézni fits nicely because watching a movie is usually thought of as a complete event.

What exactly does the meg- prefix in megnézni do?

Meg- is a perfective verb prefix. With nézni:

nézni – to watch / to look (ongoing, process-focused)
megnézni – to watch / to see (to the end, to completion)

So:

Megnézek egy filmet. – I’ll watch a movie (I’ll get through it).
Nézek egy filmet. – I’m watching a movie (right now / for some time).

In your sentence, megnézni expresses the preference for completing the action (watching a whole movie in the evening), not just looking at it for a bit.

Why is it szeretek megnézni and not szeretem megnézni?

There are two different patterns:

szeretek + infinitive
→ I like to do X / I like doing X
Example:
- Szeretek megnézni egy érdekes filmet. – I like to watch an interesting movie (in general).
szeretem + noun / specific object
→ I like / love this thing
Examples:
- Szeretem az érdekes filmeket. – I like interesting movies.
- Szeretem ezt a filmet. – I like this movie.

So:

Szeretek megnézni egy érdekes filmet. – “I like to watch an interesting movie (as an activity).”
Szeretem az érdekes filmeket. – “I like interesting movies (as a category).”

With a following infinitive (megnézni), you use szeretek, not szeretem.

Where is the “I” in this sentence? Why is én missing?

The subject “I” is expressed by the verb ending, so the pronoun én is usually dropped unless you want to emphasize it:

Szeretek megnézni egy érdekes filmet. – I like to watch an interesting movie.
Én szeretek megnézni egy érdekes filmet. – I (as opposed to others) like to watch an interesting movie.

The ending -ek in szeretek shows 1st person singular, so the subject én is understood and normally omitted.

What is the form szeretek exactly? How is it conjugated?

Szeretek comes from the verb szeretni (to like / to love).

Breakdown:

szeret- – verb stem (to like / love)
-ek – 1st person singular, indefinite present tense ending

So:

szeretek – I like / I love (indefinite)
szeretem – I like / love it / the X (definite, when there’s a specific definite object)

Examples:

Szeretek olvasni. – I like to read. (indefinite, with infinitive)
Szeretem a könyvet. – I like the book. (definite object, so definite conjugation)

Why is it filmet and not just film?

Filmet is the accusative form of film (the direct object).

Formation:

film – base form (a film/movie)
-t – accusative case ending (marks “what is watched”)
a linking vowel -e- is added: film + e + t → filmet

In Hungarian, direct objects generally take -t:

látok egy házat – I see a house
olvasok egy könyvet – I read a book
megnézek egy filmet – I watch a movie

Because egy érdekes filmet is what you watch, it must be in the accusative.

Why doesn’t érdekes change form? Shouldn’t the adjective agree with filmet?

Hungarian adjectives do not change form for case, gender, or number when they directly precede a noun (attributive position).

So:

egy érdekes film – an interesting movie
érdekes filmet – an interesting movie (accusative)
érdekes filmek – interesting movies
érdekes filmeket – interesting movies (accusative)

Only the noun changes (film → filmet, filmek, filmeket), the adjective érdekes stays the same.

Adjectives only take endings when they stand alone as predicates:

A film érdekes. – The movie is interesting.
A film érdekesnek tűnik. – The movie seems interesting.

In such predicate uses, they can take case/number endings, but not when directly before a noun.

Why do we say egy érdekes filmet? Can we drop egy?

Egy is the indefinite article (“a / an”).

egy érdekes filmet – an interesting movie (in general, not a specific one)

You can drop egy and say:

Érdekes filmet szeretek megnézni este.

This is still grammatical, but:

Including egy makes it sound more like “one whole interesting movie” as a typical unit/activity in the evening.
Without egy, it’s a bit more generic-sounding and slightly more “bare” in style.

Both are acceptable in everyday speech, but egy érdekes filmet is very natural here.

How would I say “In the evenings I like to watch interesting movies” (plural / habitual)?

You can shift both number and the time expression:

Esténként szeretek megnézni érdekes filmeket.

Breakdown:

Esténként – in the evenings / on evenings (explicitly habitual)
szeretek megnézni – I like to watch (to completion)
érdekes filmeket – interesting movies (plural accusative: filmek + -et → filmeket)

Another natural variant:

Este szeretek érdekes filmeket megnézni.
Still understandable as a habit, but esténként makes the habitual nature clearer.

Can I use nézni instead of megnézni without sounding wrong?

Yes, you can:

Este szeretek nézni egy érdekes filmet.

This is grammatically fine and understandable. The difference:

nézni – emphasizes the process of watching.
megnézni – emphasizes doing it to completion, getting through the movie.

In everyday conversation, people very often say megnézni with film, because a movie is normally viewed as a whole event. Using nézni doesn’t sound wrong, but megnézni is the more idiomatic choice here.