"'Truly I tell you, I will not drink again from the fruit of the vine until that day when I drink it new in the kingdom of God.'" NIV translation
This verse speaks of the traditions of the Passover cup. Although not specifically mentioned in the description of the Passover meal in Exodus, Jesus was talking about the four cups of the traditional seder.
According to tradition, there are four cups of wine (juice) that are served with the meal. The first cup is called the Kiddush which means sanctification and is a picture of God calling His people. The second cup is called the cup of plagues and is a reminder of the tools that God used to bring His people out of captivity. The third cup is called the cup of redemption or the cup of blessing. The final cup is called hallel which means praise but in some traditions this is called the cup of Elijah.
Jesus would have shared three cups with the disciples at this point. The first would extend the offer or calling to all people with the second being a picture of the sin that had all men in captivity. The third cup is the cup that Jesus would have described as the "blood of the covenant".
This was a bold statement that it was through His blood that men would be redeemed. There is no other way that men can be saved! Paul reminded us, in 1 Corinthians 11, that celebrating the Passover by drinking the cups without remembering that fact will bring judgment on those who do so.
In saying that He would not drink from the cup again until it was in the kingdom of heaven, Jesus was saying that they were not going to take the traditional fourth cup at this Passover seder. He was, in fact saying, that the praise would only come with the kingdom of heaven. Meanwhile, He was looking to the cup of wrath that he would take in order to pay the penalty for our sins.
"When they had sung a hymn, they went out to the Mount of Olives." NIV translation
The singing of the hymn would have probably referred to the tradition of singing Psalms 113-118 which are the traditional praise hymns. Then they went to the garden of Gethsemane.