Learn Hardcore Croatian: Vaš novi pas uči brzo u školi. - Your new dog learns quickly at school.

vaš is the possessive pronoun your in the polite/formal singular or in the plural sense (you all). It agrees with the noun pas (masculine singular nominative).
Use tvoj when addressing one person informally. Use vaš when you want to be polite, or when you are speaking to more than one person.

Why is novi in the form it is, and why is it placed between vaš and pas?

Adjectives in Croatian agree in gender, number, and case with the noun they modify. Here pas is masculine singular nominative, so novi also takes the masculine singular nominative ending -i.
Word order in Croatian is flexible, but the most neutral sequence is Possessive → Adjective → Noun, giving vaš novi pas. You could technically say novi vaš pas or vaš pas novi, but they sound less natural.

What does uči mean here, and how is it different from naučiti?

The verb učiti (imperfective) in present tense third person singular is uči, meaning “is learning.”
By contrast, naučiti is the perfective pair, focusing on completing the act of learning. Its present forms actually convey a future sense (e.g. naučit će means “will learn”). So:

uči = he/she/it is learning (ongoing action)
naučit će = he/she/it will learn (action seen as a whole)

Could uči ever mean “teaches” instead of “learns”?

Yes. učiti can also mean “to teach” when you specify a person as the object (e.g. Učim djecu engleski = I teach children English). In Vaš novi pas uči brzo u školi, the subject is the dog, so the only logical reading is that the dog is learning.

How is the adverb brzo formed, and why isn’t it brz or brzim?

Adverbs in Croatian are typically formed from the neuter nominative form of an adjective plus -o. The adjective for “fast” is brz, its neuter nominative form is brzo, which doubles as an adverb meaning “quickly.”
Forms like brz (adjective stem) or brzim (instrumental case) are not used as adverbs.

Why is it u školi (locative) and not u školu (accusative)?

The preposition u takes:

the Locative when indicating location (where?): u školi = in/at school.
the Accusative when indicating motion toward (where to?): u školu = into/to school.
Here we describe where the dog is learning, so we use the locative škola → školi.

Can I change the word order, for example Vaš pas brzo uči u školi, and will the meaning change?

Yes. Croatian word order is relatively free thanks to its case system.
• Vaš pas brzo uči u školi places a bit more emphasis on brzo (the speed).
• U školi vaš novi pas uči brzo emphasizes the location first.
The core meaning remains the same, though the neutral statement is Vaš novi pas uči brzo u školi.