We do continue on in some way after we die. Our soul (psyche), continues (Rev 6:9; 20:4). In these verses, the souls of those who died for the faith are with God in heaven.
Sheol (Hebrew) and Hades (Greek) refer to the place where dead people go. However, Hades in the New Testament can also have the sense of being a place of judgement – see Jesus’ parable of the Rich Man and Lazarus in Luke 16:19-31. In this parable, the Rich Man ends up in Hades, where he is in torment, and Lazarus ends up at Abraham’s side, with an uncrossable gulf separating them.
From these verses we also see that those who are in Christ end up in a conscious state. Those who are not in Christ, end up in torment (see also 2 Peter 2:9).
When Jesus is talking to the criminal being crucified with him, he says “Today you will be with me in Paradise”. This suggests that the movement is immediate on death. It also suggests a pleasant place. Paul also gives this impression in Philippians 1:21-23 where he says “to die is gain” and that his “desire is to depart and be with Christ, for that is far better.” In 2 Corinthians 5:1-10, he says similar things.
It does appear that we do not receive our full resurrection bodies until the end of this world (1 Corinthians 15:50-52).
When Jesus returns, those who are in Christ will be changed into our resurrection bodies. Those who are not in Christ “will be punished with everlasting destruction and shut out from the presence of the Lord” (2 Thessalonians 1:9a).