Learn Hardcore Russian: Другие люди любят шумный город.

другие люди – other people (not the ones we’ve just talked about, or not “we”).
другие can sometimes mean “different”, but in this sentence it’s more like “other” in contrast to some implied group (for example, “I don’t like it, but other people do”).
A close synonym is остальные люди – “the rest of the people”, but that sounds more like “all the others”, while другие люди can be less total and more general.

Why is it другие люди, not другой люди?

Because adjectives in Russian must agree with the noun in number, gender, and case.

люди is plural (people), grammatically masculine plural.
So the adjective другой must also be plural: другие.

Forms of другой:

Masculine singular: другой человек (another person)
Feminine singular: другая девушка (another girl)
Neuter singular: другое место (another place)
Plural: другие люди / другие места (other people / other places)

So другой люди is ungrammatical; it mixes singular adjective with plural noun.

What case are другие люди and шумный город in, and why do they look like nominative if one is the object?

Другие люди is in the nominative plural: it’s the subject of the sentence (the ones who do the liking).
Шумный город is in the accusative singular: it’s the direct object (the thing they like).

For masculine inanimate nouns, the accusative form = nominative form.
So:

Nominative: город
Accusative: город (same form, because it’s inanimate and masculine)

The adjective шумный also looks the same in nominative and accusative masculine singular for inanimate nouns:

Nominative: шумный город
Accusative: шумный город

So grammatically it’s accusative, but it looks like nominative, which is normal for this type of noun.

Why is the verb любят here? What is the infinitive and how is it conjugated?

The infinitive is любить – “to love / to like”.

In the present tense:

я люблю – I love / like
ты любишь – you (sg., informal) love / like
он / она / оно любит – he / she / it loves / likes
мы любим – we love / like
вы любите – you (pl. or formal) love / like
они любят – they love / like

In the sentence, the subject is другие люди = они (they), so we need они любят. That’s why it’s любят.

What’s the difference between любить and нравиться? Could I say Другим людям нравится шумный город instead?

Both are possible, but they work differently.

любить + accusative:
- другие люди любят шумный город – other people like / love the noisy city.
- SUBJECT is the one who likes; OBJECT is what they like.
нравиться + dative:
- другим людям нравится шумный город – the noisy city is pleasing to other people.
- Literally: “To other people, the noisy city is pleasing.”
- The thing that is liked is the grammatical subject; the person is in dative (другим людям).

Meaning-wise here, both sentences are natural, with a small nuance:

любят can sound a bit more active / emotional.
нравится is slightly more neutral “they find it nice / pleasant”.

Why is there no preposition в (“in”) before шумный город? In English I might say “like it in the noisy city.”

In Russian, любить normally takes a direct object without a preposition:

любить город – to love / like the city
любить музыку – to love / like music
любить тишину – to love / like silence

The sentence is about liking the city itself, not about being “in the city”.

If you wanted to talk about living in a noisy city, then you’d use в and a different case:

другие люди любят жить в шумном городе – other people like living in a noisy city.

Here в шумном городе (prepositional case) answers “where?”, but шумный город in the original sentence is simply the object of любят.

Why is there no word like “the” or “a” before другие люди or шумный город?

Russian has no articles (no equivalents of “a / an / the”).

The ideas of definite / indefinite (a city vs the city) are expressed by:

context
word order
sometimes demonstratives like этот (this), тот (that)

So шумный город could mean:

a noisy city (in general)
the noisy city (that we both know about)

English has to pick one; Russian doesn’t mark that grammatically in this sentence.

Why is the adjective шумный and not шумная / шумное / шумные?

Because adjectives must agree with the noun in gender, number, and case.

город is masculine singular.
In the nominative and (for inanimate) accusative singular, masculine adjectives take -ый / -ий.

So:

Masculine: шумный город (noisy city)
Feminine: шумная улица (noisy street)
Neuter: шумное место (noisy place)
Plural: шумные города / шумные улицы / шумные места (noisy cities/streets/places)

In this sentence, город is masculine singular, so шумный is the correct form.

How would the sentence change if we talked about noisy cities (plural) instead of one city?

You’d make город plural and adjust the adjective:

Другие люди любят шумные города. – Other people like noisy cities.

Changes:

город → города (nominative/accusative plural)
шумный → шумные (adjective agrees in plural)
любят stays the same because the subject другие люди is still plural.

How do I say “Other people don’t like the noisy city”? Where do I put не?

To negate the verb, put не directly before it:

Другие люди не любят шумный город. – Other people don’t like the noisy city.

Word-by-word:

другие люди – other people (subject)
не любят – do not like
шумный город – the noisy city (object)

Nothing else in the sentence changes; you just add не before любят.

Where is the stress in each word, and how should I pronounce the whole sentence naturally?

Stress (capital letters show stressed syllable):

дру́гие – dru-GI-ye (IPA: [drʊˈgʲijɪ]) – stress on -ги-
лю́ди – LYU-di (IPA: [ˈlʲʉdʲɪ]) – stress on лю́
лю́бят – LYU-byat (IPA: [ˈlʲʉbʲət]) – stress on лю́
шу́мный – SHU-mnyi (IPA: [ˈʂumnɨj]) – stress on шу́
го́род – GO-rad (IPA: [ˈgorət]) – stress on го́

Spoken smoothly, it sounds like:
дру́гие лю́ди лю́бят шу́мный го́род
Try to keep лю soft (with ю after л) and the ш in шу́мный as a strong “sh” sound.