Σήμερα πήγαμε το κατοικίδιο στον κτηνίατρο, γιατί δεν έτρωγε καλά.

In Greek, πηγαίνω / πάω can be used transitively (with a direct object) to mean take (someone/something somewhere), especially in everyday speech.
So πήγαμε το κατοικίδιο στον κτηνίατρο literally looks like “we went the pet to the vet,” but it means “we took the pet to the vet.”
A more explicit alternative is πήγαμε το κατοικίδιο μας στον κτηνίατρο (we took our pet to the vet) or το πήγαμε στον κτηνίατρο (we took it to the vet).

κατοικίδιο is neuter, so it takes the neuter article το in the singular.
Here it’s also the direct object of πήγαμε (“we took what?”), so it appears in the accusative, which for neuter looks the same as the nominative: το κατοικίδιο.

κτηνίατρος is the dictionary (nominative) form. After σε/στον, the noun must be accusative, so κτηνίατρος → κτηνίατρο.
So: ο κτηνίατρος (the vet, subject) but στον κτηνίατρο (to the vet).

Greek verb endings show the subject. πήγαμε already encodes “we” (1st person plural), so εμείς is usually unnecessary.
You’d add εμείς mainly for contrast or emphasis: Σήμερα εμείς πήγαμε το κατοικίδιο… (“we took the pet…”).

In Greek, it’s common to use a comma before γιατί when it introduces a reason clause, similar to English punctuation in many cases:
…, γιατί δεν έτρωγε καλά. = “..., because it wasn’t eating well.”

έτρωγε is imperfect, which typically describes an ongoing/repeated situation in the past.
So δεν έτρωγε καλά suggests the pet had not been eating well (for a while / repeatedly), which motivated the vet visit.
If you used aorist (δεν έφαγε καλά), it would sound more like a single completed instance (“it didn’t eat well [that time]”).

καλά is an adverb modifying έτρωγε: “eat well.”
So δεν έτρωγε καλά = “it wasn’t eating well” (not eating properly).
If you wanted “not eating much,” you’d more likely say δεν έτρωγε πολύ (“it wasn’t eating much”).