Last night, we went around the whole town drinking and partying it up.
Literal
We [topic-は] yesterday [of-の] night, town inside [object-を] drink-and horseplay do-and walk-around.
Rich narrative. 町中 ('the whole town, around town') + を marks the path traversed. 飲んでばか騒ぎをして chains three activities via te-form: 'drinking + raising hell + ...' 回る ('to go around') here attached via te-form = 'went around doing.' ばか騒ぎ literally '[fool] racket' = 'horseplay, wild partying.' A vivid description of a raucous night out.