These are instruction regarding three annual festivals
There were other feasts and sacred days, but we will cover those later, as they come up in the text.
The Feast of Unleavened Bread
During this feast, the Israelites were to eat unleavened bread (bread made without yeast) for seven days. [When hastily departing from Egypt, the Israelites had no time to put leaven in their bread and let it rise.] The time appointed for this feast was in the month of Abib, which Exodus 23:15 states was the month in which the Israelites had been delivered from Egypt. Abib corresponds to mid-March to Mid-April, and was considered the first month of the year. This feast began with the Passover meal and continued for seven days. Exodus chapter 12 explains the Passover and this Feast of Unleavened Bread quite thoroughly.
Here is just one of the significant verses in that chapter:
Feast of Harvest
This feast had Messianic implications. In New Testament times a group called the Gnostics was promoting the idea that Jesus had not actually physically risen from the dead. They rejected the idea of a future physical resurrection of the body and believed that only the soul was immortal.
The apostle Paul soon made short work of this theory. If the dead were not raised, he reasoned, then Christ would not have been raised, and thus believers ought then "to be pitied more than all men." (I Corinthians 15:19) For if Jesus had not risen, then Paul had lied to them, and their sins remained unforgiven. Also, their loved ones who had died already would have perished without hope. Not to mention that some believers had been persecuted and even martyred for their faith. All this would be in vain if Christ was not risen.
What does all this have to do with the barley harvest and firstfruits?
I'm getting there, really.
The Feast of Harvest and the Feast of Unleavened Bread points ahead to Jesus Christ, the One who would be the first to be resurrected, never to die again. [Others had been raised from the dead, like Lazarus, and Jairus' daughter, but they eventually died again at the end of their lifespans.] Just as God's acceptance of the firstfruits of the grain harvest of the Israelites was a promise that the rest of the harvest would occur, the fact that God had raised Jesus from the dead assured believers that God had accepted Jesus' death as payment for their sins and that they, too, would eventually be raised to eternal life.
The Gnostics' rejection of an actual physical resurrection was an error. Instead, Paul insisted that
"...Christ died for our sins according to the Scriptures, that he was buried, that he was raised on the third day according to the Scriptures, and that he appeared to Peter, and then to the Twelve. After that, he appeared to more than five hundred of the brothers at the same time, most of whom are still living, though some have fallen asleep. Then he appeared to James, then to all the apostles, and last of all he appeared to me also, as to one abnormally born." (I Corinthians 15:3-8 NIV)
Paul gives a list of people who had seen the risen Christ. Most were still alive, so doubters could go check with them in person, if they wanted to do so. Although we do not have that same privilege, it is worth noting for our own consideration of these matters that many of these witnesses -- and all of the apostles, except John -- later died for their faith, a fact which would seem unlikely if Jesus' resurrection had not occurred.
The Feast of Ingathering
The Feast of Ingathering was celebrated at the end of the agricultural year [about mid-September to mid-October] when crops were gathered from the fields. It was also sometimes called the Feast of Tabernacles or Feast of Booths because the Israelites had lived in temporary shelters during their journey out of Egypt. This Feast commemorated the desert wanderings after the exodus from Egypt. As the Israelites gathered their crops, they would probably think back upon how God had provided for them during their wandering, and rejoice in His continued provision.
Three time a year all the men were to appear before the Lord. If they offered a sacrifice, its blood was not to be offered with anything containing yeast. Yeast was often used in the Bible as a symbol for sin. These feasts often contained certain aspects which pointed ahead to the greatest sacrifice, when the Messiah Jesus Christ offered Himself as the propitiation for the sins of the world. In order for him to be able to do so, He had to be sinless, and thus it would be necessary for any symbol of this sacrifice also to be free of any taint of sin.
The best of the Israelites' firstfruits were to be offered to the LORD. For me, this recalls the sacrifices offered by Cain and Abel: Abel offered the best from his flocks, while Cain offered "some" of the produce which he had grown (Genesis 4:1-5). Abel's offering was accepted, while Cain's was rejected.
A young goat was not to be cooked in its mother's milk. This may have been forbidden because doing so was similar to some ritual among the nations surrounding Israel in their worship of other 'gods'. Avoiding such a practice could guard against syncretism (combining elements from several belief systems into one). Or there may be an inherent repugnance regarding this because it seems to violate the idea of what the mother-child relationship should be -- one of safety, provision, comfort, and protection.