Because he/she finished their final round of vaccines, ~ went on their first walk today!
「犬のワクチン」は英語で「Dog vaccine」という意味がありますが、「ポチは先日3回目のワクチンを打ち終わったので今日、お散歩デビューしました。」には「Pochi has debuted on his first walk today, after completing his 3rd round of vaccinations!」という表現が良いと思います。
先日ーThe other day・A few days ago
三回目のワクチンー3rd round of vaccinations
打ち終わるーTo finish getting a vaccine (ワクチンを打つーTo take a vaccine・To get a vaccine)
お散歩デビュー-First walk・Debut walk