So why should we think that the first day of both the spring Feast of Unleavened Bread and the fall Feast of Tabernacles is the seventh-day Sabbath?
Answer: We cannot separate Hebrew calendation from the Hebrew religion, because they were both given by Yahuwah on several occasions. At creation the dual action of the sun and moon was divinely appointed by Yahuwah to rule in the heavens. These two were to rule together over signs and seasons, and over days and years. See Genesis 1:14-18. But what exactly is meant by the signs and seasons? Signs are new moons and Sabbaths complete; while “seasons” actually come from the Hebrew word mo’ed. These are Yahuwah’s appointed annual feasts and not the four quarters of the Roman year.
A full 2500 years after creation Yahuwah restored His calendation to Moses and the Hebrew people. Remember it was then that Moses recorded both the Genesis account as well as the Exodus and Leviticus account. Both time systems from Genesis and the Exodus had to agree or Moses would have recognized a discrepancy. This is one of the checks and balances. These first three months beginning in Exodus 12 are three identical months in a row. This is not possible on the Gregorian calendar.
So to answer your question -- we can know that the first day of the Feast of Unleavened Bread and Feast of Tabernacles was the seventh-day Sabbath because the dates are given that line up with the moon phases. The same details that were given at the Exodus were perfectly fulfilled by Yahushua 1500 years later. You do believe that He was crucified on the 6th day of the week, rested in the tomb on the seventh-day Sabbath, which was the first day of The Feast of Unleavened Bread? It was because this Feast began on a seventh-day Sabbath that made it a High Sabbath. These two High Sabbaths occur each and every year; they are always the first day of the Feast of Unleavened Bread and the Feast of Tabernacles. Because the Feast of Weeks (Pentecost) is not a fixed date, it will sometimes also coincide with a weekly Sabbath.
It is impossible to take the details for Yahuwah’s Feasts as given in Leviticus 23 and apply them to the Gregorian calendar. The reason is the two simply can not be harmonized. Try it and prove it to yourself that for just one year it won’t work. If Yahuwah’s holy Feasts cannot be harmonized for just six months then isn’t it proof enough that the Gregorian calendar is simply not the calendar of the Scriptures.