Learn Hardcore Croatian: Želim staviti novu fotografiju na mrežu. - I want to put a new photo on the internet.

Croatian usually drops subject pronouns because the verb ending already shows who the subject is.
Želim is the 1st person singular form of željeti (to want), so it can only mean I want.
You add ja (Ja želim staviti…) only for emphasis, for example to contrast with someone else: Ja želim staviti fotografiju, a ti ne želiš.

What’s the difference between želim and hoću? Both seem to mean I want.

Both želim and hoću can translate as I want, but there is a nuance:

Želim (from željeti) is a bit more polite or neutral, often close to I’d like.
Hoću (from htjeti) can sound more direct or demanding, especially in some contexts.

In this sentence, Želim staviti novu fotografiju na mrežu sounds perfectly neutral and natural.

Why is the verb staviti in the infinitive? Could it be stavljati instead?

After želim, Croatian normally uses the infinitive: želim + infinitive.
Staviti is a perfective verb, meaning to put once / to place (as a single completed action).
If you said Želim stavljati nove fotografije na mrežu, using stavljati (imperfective), it would mean I want to keep putting / regularly put new photos online, focusing on repeated or ongoing action.
With staviti, the sentence talks about one specific act of putting a new photo online.

What case is novu fotografiju in, and why?

Novu fotografiju is in the accusative singular, feminine.
The verb staviti takes a direct object in the accusative (what are you putting? → a new photo).
Base form (nominative): nova fotografija.
Accusative: novu fotografiju → the noun fotografija becomes fotografiju, and the adjective nova becomes novu to match it.

Why does nova change to novu in novu fotografiju?

Adjectives in Croatian must agree with the noun in gender, number, and case.

Nominative feminine singular: nova fotografija (-a, -a)
Accusative feminine singular: novu fotografiju (-u, -u)

Since fotografiju is accusative feminine singular, nova must also take the accusative feminine singular form novu.

Why is it mrežu, not mreža, in na mrežu?

Here mrežu is also in the accusative singular (feminine).
The preposition na can take accusative or locative:

na + accusative → movement onto something (direction)
na + locative → position on something (location)

In staviti … na mrežu, there is movement onto the network, so na uses the accusative: mrežu.

What is the difference between na mrežu and na mreži?

Na mrežu (accusative) = onto the network / onto the web (movement, putting it there).
Na mreži (locative) = on the network / on the web (already there, location).

So you say:

Želim staviti novu fotografiju na mrežu. – I want to put it onto the web.
Fotografija je na mreži. – The photo is on the web (already uploaded).

What exactly does mreža mean here? Is it really internet?

Literally, mreža means network or net.
In this sentence, na mrežu is understood as onto the (online) network, so the natural English translation is online or on the internet.
In everyday Croatian, people might also say na internet, na društvene mreže (onto social networks), or name a specific platform (na Instagram, na Facebook).

Why do we use na and not u in this sentence?

In Croatian, na is typically used for the idea of being on / onto a surface, medium, or platform (e.g. na internet, na televiziji, na zidu).
U is more like in / into, used for enclosed spaces or groups (e.g. u kući, u gradu, u knjizi).
Since we conceptualize the internet as a surface or platform you put things on, na mrežu (onto the network) is used, not u mrežu.

Is the word order fixed? Could I say Želim novu fotografiju staviti na mrežu?

Croatian word order is quite flexible, and your version is grammatically correct: Želim novu fotografiju staviti na mrežu.
The neutral, most common order is the original: Želim staviti novu fotografiju na mrežu.
Moving novu fotografiju earlier can slightly highlight the new photo rather than the action, but in everyday speech both orders sound natural.

Are there other common ways to say staviti novu fotografiju na mrežu?

Yes, several alternatives are common in everyday speech:

Želim objaviti novu fotografiju na internetu. – I want to publish/post a new photo on the internet.
Želim postaviti novu fotografiju na internet. – I want to post/upload a new photo on the internet.
Colloquial: Želim uploadati novu fotku na Facebook/Instagram. – I want to upload a new photo/pic on Facebook/Instagram.

All of these keep the same basic idea, with slightly different levels of formality and modern/colloquial flavor.

What is the difference between fotografija and slika here?

Both fotografija and slika can mean picture in everyday language.

Fotografija is a bit more precise: specifically a photograph.
Slika is more general: a picture, image, also painting/drawing.

In this context, you could also say Želim staviti novu sliku na mrežu, which sounds a bit more casual and very common in speech.