Learn Hardcore Croatian: Ana želi hodati kroz park, ali Luka želi voziti do grada. - Ana wants to walk through the park, but Luka wants to drive to the city.

After verbs like željeti (want), morati (must), moći (can), standard Croatian prefers a bare infinitive: Ana želi hodati.
The da + present form (da hoda) is common in Serbian and in some colloquial Croatian, but in standard Croatian it's less preferred, especially after those verbs.
So the given sentence is the standard choice.

What’s the difference between hodati, šetati, and ići pješice?

hodati: to walk (general manner of movement; neutral).
šetati: to stroll, walk for pleasure; sounds very natural with parks: šetati (kroz) park or šetati po parku.
ići pješice: to go on foot to a destination: ići pješice do grada.

Does hodati ever mean “to date” someone?

Yes. hodati s nekim = to be dating someone: Ana hoda s Lukom.
For literal walking, add a place or use šetati to avoid ambiguity: hodati kroz park, šetati s Lukom kroz park.

Which case does kroz take, and what form is park in?

kroz takes the accusative.
park is a masculine inanimate noun; its accusative equals the nominative, so kroz park is correct.
Example with a change: kroz šumu (from šuma → accusative šumu).

Why is it do grada and not do grad?

Can I say u grad instead of do grada? What’s the nuance?

u grad (+ accusative) = into the city, highlighting entry: voziti u grad.
do grada (+ genitive) = up to/to the city as an endpoint. In practice it’s also used for reaching the city.
Both are common; use u grad if you want to stress going into the city proper.

Does voziti mean “to drive” or “to ride”?

voziti: to drive/operate a vehicle, or to give someone a ride: Vozim auto. Vozim Anu do grada.
voziti se: to ride/travel by vehicle (you may be a passenger): Luka se želi voziti do grada.

Where does se go in a sentence like this?

Could I use hoće instead of želi?

Yes. htjeti (present: hoću, hoćeš, hoće, …) also means to want.
želi is neutral/polite; hoće can sound more direct or blunt. Both are fine: Luka hoće voziti do grada.

Is a the same as ali here? What about nego?

ali = but/however, stronger contrast: … ali Luka želi …
a = and/but with milder contrast or topic shift: … a Luka želi … (also correct here).
After a negation, use nego for a corrective contrast: Ne želi hodati, nego voziti.

Why is there a comma before ali?

Croatian uses a comma before coordinating conjunctions like ali, a, nego when they connect independent clauses. Both parts are full clauses here, so the comma is required.

Where are the articles? How do I say the park or a park?

Croatian has no articles. park can mean either the park or a park, depending on context.
To be specific, use demonstratives: taj park (that/the park), ovaj park (this park). For an indefinite feel, jedan park can be used.

How flexible is the word order?

Quite flexible for emphasis; the given order is neutral.
Possible variants:
- Ana želi kroz park hodati (highlights the path).
- Kroz park Ana želi hodati (fronts the location).
- Avoid overly marked orders like Luka do grada želi voziti unless for special emphasis.

Any pronunciation tips for this sentence?

ž (in želi) is like English “zh” in vision.
r is trilled; u (in Luka) is like oo in food.
In fast speech, across kroz park, the final z often devoices, so you may hear something like kros park.

Are the names Ana and Luka declined in cases?

Yes.
Ana: Gen Ane, Dat/Loc Ani, Acc Anu, Instr Anom.
Luka (masculine): Gen Luke, Dat/Loc Luki, Acc Luku, Instr Lukom. Vocative is usually the same as nominative: Luka!

Could I say prošetati instead of hodati/šetati? What’s the aspect difference?

šetati is imperfective (ongoing/habitual/leisurely activity).
prošetati is perfective (a single, bounded stroll).
So Ana želi šetati kroz park = she wants to stroll (in general/for some time), while Ana želi prošetati kroz park = she wants to go for a (one-off) stroll.