His classic studies made a little puzzle, Because of the filthy loves of gods and goddesses, Who in the earlier days raised a bustle, But never put on pantaloons or bodices (1.41)
Don Juan's mother isn't sure if she wants her son reading the classics of Western literature because the classics are often filled with sex and violence. Little does she know that by shielding her son from sexual knowledge, she's making him more likely to act on his budding desires.
Here ends this canto. —Need I sing, or say, How Juan, naked, favoured by the night, Who favours what she should not, found his way, And reached his home in an unseemly plight? (1.188)
This classic scene shows Don Juan escaping naked into the night after Alfonso has found him (Juan) with his wife, Julia. Juan is obviously naked because he and Julia have been having adulterous sex.
Haidée and Juan were not married, but The fault was theirs, not mine: it is not fair, Chaste reader, then, in any way to put The blame on me, unless you wish they were (3.12)
Byron admits that he has been writing about a sexual relationship where Juan and Haidée are not married, which would have been immoral during his time. But here he asks the audience not to blame him, since it was Don Juan and Haidée who made the decision to have sex.
In her first passion Woman loves her lover, In all the others all she loves is Love, Which grows a habit she can ne'er get over, And fits her loosely—like an easy glove (3.3)
Byron suggests that the first time a woman has sex, she is totally in love with the person she's having sex with. From that point on though, the woman loves the idea of being in love more than whatever specific man she's with. Patronizing much, Byron?
The favour of the Empress was agreeable; And though the duty waxed a little hard, Young people at his time of life should be able To come off handsomely in that regard. (10.22)
Earmuffs, all you youngsters out there. Byron's use of the word "hard" here could humorously refer to Don Juan's erection. On top of that, Byron jokingly suggests that the best way for a young man to make money with an Empress is to turn himself into a prostitute.
One monstrous diamond […] Bestowed upon him, as the public learned; And, to say truth, it had been fairly earned (11.39)
Byron hints that Don Juan has earned an expensive gift from Catherine the Great by being a sort of sex servant to her. He never confirms this one way or the other, but that's exactly what makes it so titillating.