Sexuality in the Original Series echoes many of the sentiments of classic science fiction of the early to late twentieth-century, particularly the late 1970s, when the series first aired.
Such relationships were purely heterosexual in nature, with an emphasis of certain males (i.e. Starbuck, Sire Uri, and Count Iblis) seeking the partnership of various females, but typically independent (and unbeknownst) to the other females.
Polygamy and Monogamy
Both polygamy and monogamy are known to the Colonials. Polygamy itself is a mating practice that dates as far back as the Lords of Kobol with each lord having multiple wives in a time before the exodus from Kobol. Such wives were typically "very young" according to the Book of the Word (TOS: "The Living Legend, Part I"). It remains unstated whether or not bisexual sex acts were involved in these polygamous relationships.
Yet despite this, monogamy appears to be the default mating practice of the Colonies, as many Colonials were Sealed to only one mate, among them:
Although monogamy apparently reigns on the Colonies, many Colonial males such as Starbuck, Chameleon, and Sire Uri tend to regale themselves with multiple female companions.
Being the primary "swash-bucking" character, Starbuck tends to engage in flirtatious relationships with the various women he has come across, including: Athena, Aurora (TOS: "Take the Celestra"), Cassiopeia, Miri (TOS: "The Young Lords"), and even attempted to flirt with Sheba (TOS: "The Living Legend, Part I").
In addition, it is indicated that women also regaled themselves with multiple male companions. In "War of the Gods, Part II", Iblis comments to Starbuck that Sheba should not limit herself to one man, when Starbuck notes that Sheba seemed to dump Iblis in favor of Apollo during a Fleet celebration.
Also, sexual relationships seem to be fairly fancy-free, as Siress Blassie takes a sexual interest in Chameleon, who arranges for his affairs after it was "proven" that Chameleon was not Starbuck's father (TOS: "The Man with Nine Lives").
Socialators are a form of sexual companion serving the Colonies. Their vocation is sanctified by "the elders", whose gender was never directly specified either way. People practicing this vocation are viewed with disdain by religious groups like the Otori Sect and, judging by Cassiopeia's quick defense of the vocation to Starbuck, others (TOS: "Saga of a Star World").
The most religious of the Twelve Colonies, the Gemonese, treat contact between the sexes very differently than members from other Colonies. The aforementioned Otori Sect's high priests permit contact between the genders during the Worship of the Sun Storm, which occurs every seven yahren (TOS: "Saga of a Star World").