There is one verse that describes a spirit going to heaven when we die:
… and the dust returns to the earth as it was, and the spirit returns to God who gave it. (Ecclesiastes 12:7)
This is an allusion back to Genesis 2:7 and the creation of Adam.
He was made from dust and God breathed into him the “breath of life” to make him a living person. The word “breath” is the same as the word “spirit” in Hebrew. When we die, the life-force that keeps us alive returns to God. He keeps all things alive by his power or spirit. This is not a conscious part of us, it is simply the power of God that keeps us alive. Psalm 104:29-30 says much the same thing about animals.
There is nothing in scripture about us having any kind of consciousness while dead (apart from a few highly allegorical poems in the prophets). In fact, several verses teach quite clearly that there is no consciousness in death. See Ecclesiastes 9:5; Psalm 6:5; 115:17.
The Bible does describe the hope of life after death, but it is on earth and not in heaven, and it happens when Jesus returns, not the moment we die.
See After death what? for more information.