After that she'd go get drunk three times a week, bringing home a different man each time.
Literal
That [gen-の] after [also-も] week 3-times [gen-の] pace [with-で] get-drunk, each-time different man [obj-を] bring-come.
週三回のペース 'a pace of three times a week' uses ~ペース, a productive pattern for describing frequency. そのつど 'each time / every such occasion' (also written その都度) is a set adverbial phrase. 連れてくる 'to bring along (toward speaker)' is a compound benefactive motion verb — 連れる 'accompany' + くる 'come' (the motion-toward-speaker sense of ~てくる). The continuous casual narrative present tense describes a habitual past pattern.