According to Calvinism, when and what happen during the Lord's Passover in the time of Moses?

I don't think Calvin said anything directly but Matthew Poole, from the Reformed Tradition says in his commentary:

And ye shall keep it up until the fourteenth day of the same month: and the whole assembly of the congregation of Israel shall kill it in the evening. Ye shall keep it up; separate it from the rest of the flock, and keep it in a safe place.

In the evening; Heb. between the evenings, or the two evenings, i.e. between the beginning and end of the evening. The evening is one third part of the day, and one of the appointed and usual times of devotion, as appears from Psalm 55:17 Daniel 6:10; and it begun at their ninth or our third hour, as may be gathered from Acts 3:1; for then the sun began more sensibly to decline, whence that time is fitly called by the Jews the first evening, and that was the time of the evening sacrifice; the second evening was when the sun was setting or set. Between these it was to be killed.

This view is very commonly held. The Lamb was slain on the afternoon of the fourteenth day of the month.

It then had to be roasted and was eaten, a few hours after it had been killed, that same evening after sunset.

(Once it was dark the date changed to fifteenth.)

On the night of the first Passover the first-born sons of the Egyptians died at midnight, and the Israelites were driven out. They ate the meal fully girded, that is to say dressed and ready for the sudden departure which they made.

Calvin's view of the 14th or 15th of Nisan observance is inferred, rather than explicit, but there's no reason not to conclude that Calvin thought Christ died on the 15th of Nisan a Passover Friday.

He believes that Christ ate the Passover on Thursday night with that night being the start of the 15th with Friday day the 15th until sunset being the day of crucifixion. The Jews, he thought, would postpone their Passover by one day to the 16th a weekly Sabbath. This is so such that they would not observe two non-working days (Sabbaths) in a row. This view assumes (infers) that the first day of Unleavened Bread, the 15th, was considered a feast Sabbath.

As well, Calvin explicitly believes that Christ never violated the Law. This too infers that Calvin thought the initial Mosaic observance of Passover occurred on the 15th.

So, in Calvin's view Friday the 15th Passover was a Sabbath and on which Christ died. It was followed by the 16th weekly Sabbath when the others observed Passover.

From Calvin's Harmony of the Gospels

But hence arises a more difficult question. How did Christ observe that ceremony on the day before the whole nation celebrated the public passover? For John plainly affirms that the day on which Christ was crucified was, among the Jews, the preparation, not of the Sabbath, but of the passover....

First, it is beyond a doubt that Christ was put to death on the day before the Sabbath; for he was hastily buried before sunset in a sepulcher which was at hand, (John 19:42,) because it was necessary to abstain from work after the commencement of the evening. Now it is universally admitted that, by an ancient custom, when the passover and other festivals happened on Friday, they were delayed till the following day, because the people would have reckoned it hard to abstain from work on two successive days....

Even the Jews themselves unquestionably will not deny that, whenever the Sabbath immediately followed the passover, it was on one day, instead of both, that they abstained from work, and that this was enjoined by the Rabbins. Hence it follows that Christ, in departing from the ordinary custom, attempted nothing contrary to the Law.

This view incidentally that Friday was the 15th of Nisan is traceable through the Roman Catholic Church back to about 100-200 years before Christ's time, though not all the way back to the first in Exodus. So, it is not accurate to state that the alternate view that Christ died on Passover proper the 14th is not Christian. There are historically numerous Christians (Polycarp, Polycrates, Clement of Alexandria, etc) who taught this accurate understanding of the day and date of Christ's death and burial.

As to answering how the first Passover was observed in the time of Moses, the commandment was to slay the Passover at sunset between the evenings as the 13th ended and the 14th began. They would eat it on the 14th at night. The death angel would come through at midnight of the 14th. In the 14th at day, they gathered at Ramses. They left Egypt at night on the 15th the morrow of the Passover (Num. 33:3).