Learn Farsi: اون میره کافه. - He's going to the cafe.

In spoken Farsi, the preposition به (to) is very often dropped before destinations. While you would write او به کافه می‌رود in formal written Farsi, in everyday speech it is perfectly natural to leave the preposition out completely.

Shouldn't the verb go at the very end of the sentence?

Usually, yes! Farsi is generally a Subject-Object-Verb language. However, in casual spoken Farsi, it is extremely common to move a destination to the end of the sentence, right after the verb. Saying اون کافه میره is also correct, but اون میره کافه is how people actually speak in relaxed situations.

How exactly does the verb میره connect to the infinitive رفتن?

میره is the colloquial pronunciation of می‌رود. The present stem of رفتن (to go) is رو. In formal Farsi, you add the continuous prefix می and the third-person ending د to get می‌رود. In spoken Farsi, the formal ending softens to an e sound and the middle consonant is dropped, turning it into میره.

Does اون strictly mean he, or can it mean she as well?

It can mean both! Farsi pronouns do not specify gender. اون is simply the spoken form of both او (he/she) and آن (that/it). You just rely on the context of the conversation to know exactly who is going to the cafe.