While Moses was on the mountain the people were impatient and built a golden calf to worship. When Moses returned from the mountain with the two tablets of the Testimony, he questioned his brother about this, Aaron replied with the lamest excuse on record. “…they gave me this gold, and I threw it into the fire, and out came this calf!” HA!
This is a classic example of transferring blame.

We wonder why the people would so quickly dismiss all the miracles God had already done for them. We sometimes forget that the people were accustomed to seeing and hearing about the multiple gods of the Egyptians and the practices associated with their worship of the false gods.  They were copying the Egyptians when they celebrated, worshiped and honored their Egyptian gods by participating in revelry. The Hebrew word for revelry means "caressing, entertaining, jesting, laughing, making sport, mocking, playing" or sexual activity.
Exodus 32:6
Early the next morning they brought some animals to burn as sacrifices and others to eat as fellowship offerings. The people sat down to a feast, which turned into an orgy of drinking and sex. GNB

When Aaron proclaimed a "feast to the Lord" in honor of the golden calf, he was compromising and justifying the worship of a pagan god, by calling it a "feast to the Lord".  He had instituted a man-made festival. He was trying to worship God on an altar of paganism.  At this point God had not given the Israelites all the commands of the Feast and Festival Days.  Today, we know about all the God-ordained Feast and Festival Days and yet we still worship God on the altar of paganism.  God's ordained days are Feast of Unleavened Bread (Passover) and Feast of Tabernacles (birth of Jesus) and yet we in the western Christian church honor and celebrate Easter and Christmas, both with pagan origins.  And we ignore the holy days. Fat Tuesday (Mardi Gras) is an example today reminiscent of the revelry that the people indulged in as recorded in today's reading (Exodus 32).

Exodus 31:13  

"Say to the Israelites, 'Be sure to observe my days of worship. This will be a sign between me and you for generations to come so that you will know that I am the LORD who makes you holy.

We may be tempted to think that if we go to church on Sunday and don’t do any work at all on that day of the week, this is what will make us somehow righteous and devout.  These days of worship referred to in this passage certainly meant the seventh day of rest, but also the other days of worship, His holy days, the feast and festival days of Tabernacles, Passover, and Pentecost.  Worshiping Him and honoring Him on those days, would be a sign for all to see, a distinguishing mark.  This is what God says will set us apart from all the other people  on the face of the earth.  When we are set apart in this way, He says we will know (and other people will know) that He is the Lord who makes us holy.  Did you get the significance of that verse?  If we observe His days of worship, we will show the world that we are distinguished, we are set apart, we are different from the world and are made holy.

The Levites make a bold stand for the Lord.  The 3,000 people who were killed...were these the ones who were directly involved in the making of the golden calf and/or the ones leading the revelry?  The people were struck with a plague because of what they had done with the calf.  Could this be the result of heavy metal poisoning from being made to drink the water sprinkled with the dust from the golden calf?

In response to their sin, Moses makes an almost unbelievable plea on behalf of the people.
“…please forgive their sin - but if not, then blot me out of the book you have written.”
What book? (See Philippians 4:3, Psalm 69:28, Revelation 3:5, Revelation 20:12)

Moses inquires about who will go with him to lead the people into the land of promise.  The following verses are some of the most powerful in Scripture and so applicable to today.
Exodus 33:14-17
The LORD said, "My Presence will go with you and I will give you rest." Then Moses replied, "If your Presence does not go with us, do not send us up from here.  How will anyone know that you are pleased with me and with your people unless you go with us? What else will distinguish me and your people from all the other people on the face of the earth?" And the LORD said to Moses, "I will do the very thing you have asked because I am pleased with you and I know you by name."

Indeed!  The only thing to distinguish us as His people, the only thing that will set us apart from all other people, is if we have the Presence of the Holy Spirit within us!

There are some "apparent contradictions" in today's reading. God tells Moses that no one can see His face and live and yet earlier it says the Lord would speak to Moses face to face as a man speaks with a friend. The only explanation is that Moses spoke with God the Son, the preincarnate Jesus.  God, the Father is surrounded by unapproachable light.

1 Timothy 6:16
who alone has immortality, who dwells in unapproachable light, whom no one has ever seen or can see. To him be honor and eternal dominion. Amen. 

Moses chiseled out two stone tablets to take back up the mountain so that God could write on them the words that were on the first tablets which Moses broke in anger.  Moses wrote down words that God spoke to him in addition to the Ten Commandments.  The words of the Commandments were written by the finger of God. God tells Moses that the people are to obey Him and He will drive out the inhabitants from the land.  But, He gave a warning against making treaties with the people there.  In fact God tells Moses to destroy any remnant of their pagan worship because to leave anything pagan in the land will lead the Israelites into idol worship.

Exodus 34:33-35
When Moses finished speaking to them, he put a veil over his face. But whenever he entered the LORD's presence to speak with him, he removed the veil until he came out. And when he came out and told the Israelites what he had been commanded, they saw that his face was radiant. Then Moses would put the veil back over his face until he went in to speak with the LORD.
2 Corinthians 3:18
And we, who with unveiled faces all reflect the Lord's glory, are being transformed into his likeness with ever-increasing glory, which comes from the Lord, who is the Spirit.

Are our faces radiant after we have spoken with the Lord?