Why is today a weekday and not a weekend? i.e. When did this 7 day cycle start (and who started it?)?

This stretches very far back into antiquity and is based on the 7 planets known at the time" Sun, Moon, Mercury, Earth, Venus, Mars, Jupiter and Saturn. People thought the planets were gods, so they named the days of the week after them. We still have Sun's day, Moon's day, Tyr's day (Tuesday, named for the Norse god of war), Woden's Day for Wednesday, named after the Germanic god Woden, Thor's Day for Thursday, and Frigga's day for Friday. Wednesday is Mercury's day, Thor's day is Jupiter's day, and Friday is the day of Venus. Early months were lunar months of 28 days divided into 4 weeks of 7 days each.

The lunar method of telling time probably goes back to the new stone age.

The cycle of the Moon takes 28 days, and divides neatly into quarters of one week each. That's where the seven days comes from. Then the Church came along and picked Sunday as a day of worship, and then it was all downhill from there.

Counting from the new moon, the Babylonians celebrated the 7th, 14th, 21st, and 28th as "holy-days", also called "evil days" (meaning "unsuitable" for prohibited activities). On these days officials were prohibited from various activities and common men were forbidden to "make a wish", and at least the 28th was known as a "rest-day". On each of them, offerings were made to a different god and goddess. They were using a 7 day week before the 4th century BC