On minor fast days, we don't eat or drink. But my experience is that in other regards we mostly go about our usual activities -- go to work, tend to household chores, etc. What other activities are forbidden on minor fasts?
Orach Chayim 568:12 states:
כל השרוי בתענית בין שהיה מתענה על צרתו או על חלומו בין שהיה מתענה עם הצבור על צרתם הרי זה לא ינהג עידונין בעצמו ולא יקל ראשו ולא יהיה שמח וטוב לב אלא דואג ואונן כענין שנאמר מה יתאונן אדם חי
Anyone involved in fasting — whether because of his suffering, or because of his dream, or whether he's fasting with the community for their suffering — should not pamper himself, nor be lightheaded, nor be happy and lighthearted. Rather, he should worry and mourn — as it says: "What should a living man mourn?"
Clearly then, entertainment should certainly be minimised if not outright forbidden.
Along these lines, Rav Ovadiah Yosef held that listening to music was forbidden. He permitted shaving and haircutting, though, since he considered these to be chores. I haven't had the chance to find this ruling myself yet, but it is referenced here with additional details.
Orach Chayyim 550:2 states:
צומות הללו, חוץ מתשעה באב, מותרים ברחיצה וסיכה ונעילת הסנדל ותשמיש המטה.
On these fasts, except Tish'ah B'Av, one may wash oneself, anoint oneself, wear shoes, and have marital relations.
(Translation from Sefaria)
However, Mishna Berurah ad loc. adds:
ובעל נפש יחמיר בכולן כמו בט"ב אלא שאם חל ליל טבילה יקיים עונתו וגם בנעילת הסנדל לא יחמיר משום חוכא וטלולה וכן אם חל י' בטבת בע"ש אין להחמיר ברחיצה בחמין מפני כבוד השבת.
[A] pious person should be stringent on all of them as on Tish'ah Be'Av. But, if one of them falls on the evening when [a man's wife is due to] immerse [after her period], he should fulfill his obligation [to have marital relations]. Likewise, a person should not be strict about wearing shoes, in order not to be a laughingstock. So, too, if the 10th of Tevet falls on Friday, a person should not be strict about not washing in hot water, out of respect for Shabbat.
(Translation from Sefaria; emphasis mine)
Thus, according to Mishna Berurah, a pious person (בעל נפש) should refrain on a minor fast from washing and annointing themselves, and from marital relations (with certain exceptions).
[I wonder whether nowadays he would encourage such a person to be strict regarding not wearing leather footwear as well. I can think of many scenarios where not wearing leather shoes would no longer cause one to be a laughingstock.]
Whether an individual ought to consider themselves a pious person is probably a subject best discussed with one's rabbinic advisor/mentor.
