Learn Croatian: Sviđa mi se Markova nova klinika.

To say 'Marko's', we turn his name into a possessive adjective. For masculine names ending in a consonant or -o, we add the suffix -ov. Then, we have to match the gender of the thing he owns. Since klinika is feminine, we add the feminine ending -a, giving us Markova.

In English, 'clinic' is the object we like, so why are Markova nova klinika in the nominative (subject) case here?

The verb sviđati se literally translates to 'is pleasing to'. So, the Croatian sentence structure is actually 'Marko's new clinic is pleasing to me.' The clinic is doing the action of being pleasing, making it the subject in the nominative case.

I see mi (to me) and se (reflexive). Why are they in that specific order?

They are both clitics (short, unstressed words) and must follow a strict placement in the sentence. The rule for clitics is that dative pronouns like mi always come before the reflexive pronoun se.

If Marko had a new car (auto, masculine) instead of a clinic, how would the possessive adjective change?

Possessive adjectives act just like regular adjectives, so they must match the noun they describe in gender, number, and case. For a masculine noun like auto, you would drop the feminine -a and use the masculine nominative form Markov. The sentence would become: Sviđa mi se Markov novi auto.