"In that day you will say: 'I will praise you, O Lord. Although you were angry with me, your anger has turned away and you have comforted me.'" NIV translation
"In that day" refers to the kingdom described in chapter eleven as well as in Revelation 20. We see that, in this kingdom, God's people will be praising Him for His mercy.
"Surely God is my salvation; I will trust and not be afraid. The Lord, the Lord, is my strength and my song; he has become my salvation." NIV translation
Isaiah was talking to the people of Judah at this time and speaking of a time when many of them would turn away from their own efforts at salvation and accept the free gift of Jesus Christ as their Messiah. Jesus is the only way of salvation and all of us who are His will be singing this song.
"With joy you will draw water from the wells of salvation." NIV translation
Today, many of us take water for granted as we simply open the tap and water comes into our home that we can drink. The people of Judah did not have it so easy as they had to draw water from a well that was often far from their home. Imagine the joy that they would have in water coming to them instead of them having to go and carry it. Throughout the Bible water is a picture of the Spirit of God and it is this Spirit that dwells inside of us and brings forth joy.
"In that day you will say: 'Give thanks to the Lord, call on his name; make known among the nations what he has done, and proclaim that his name is exalted.'" NIV translation
Now, we see the other song and its purpose is to tell those that do not know Him about Jesus and what He has done for us and them.
Isaiah 12:5 & 6
"'Sing to the Lord, for he has done glorious things; let this be known to all the world. Shout aloud and sing for joy, people of Zion, for great is the Holy One of Israel among you.'" NIV translation
Joy; that is what the children of God should be known for. Whether it is in good times or bad, the world can and should see the joy of our Lord through us. We can be unhappy because of our circumstances but our joy is in the Lord and therefore does not depend on what is going on around us.