Learn Farsi: خب، من می‌رم خونه.

In spoken Tehrani Farsi, the present stem of the verb "to go" (رو) drops its و sound. So, the formal می‌روم (mi-rav-am) shrinks down to the highly common colloquial form می‌رم (miram).

I learned the word for house or home as خانه. Why is it spelled خونه here?

This reflects a standard colloquial Tehrani vowel shift. In everyday spoken Farsi, the long "a" sound frequently changes to an "oo" sound when it comes right before an ن (n) or م (m). As a result, خانه (khaneh) becomes خونه (khuneh).

Doesn't the verb "to go" require the preposition به (to) before the destination?

In formal written Farsi, yes, you would usually see به خانه می‌روم. However, in everyday spoken Farsi, the preposition به is almost always dropped before a destination. Just like English speakers say "I'm going home" rather than "going to home," Farsi speakers simply say من می‌رم خونه.

What exactly does خب mean at the start of this sentence?

خب (khob) is a shortened, colloquial form of the word for "good" (خوب), but at the beginning of a sentence, it functions exactly like the English conversational filler "well" or "so." Here, the speaker is using it to smoothly transition the thought, likely wrapping up a conversation before leaving.