Matthew 16:24-25 At that time Jesus said to his students, "If anyone wishes to come after me, they shall deny themself and they shall take up their cross, and they shall follow me. But whoever wishes to save their life, will lose it; but whoever loses their life for my sake, will find it."
What could it mean for Jesus' students to hear Jesus speak these words to them? How would they have heard it? The context of all three versions of this saying in the gospel come in terms of the world's acceptance in contrast to confession of Christ and acceptance by God. So there is a clear call here to be willing to identify with Jesus, even though this is means rejection by the world. More than that, 1st century Palestinian Judeans under Roman authority knew that "take up their cross" must mean "die a criminal's death at the hands of the Roman authorities". There is no abstract metaphor here, no "bearing life's sufferings", or "my cross to bear". For the twelve, hearing Jesus' words, siding with Jesus means a resolution to face criminal and/or political execution.
Mark 8:34-35 And calling the crowd with his students, he said to them, "if anyone wishes to follow after me, they shall deny themself and they shall take up their cross, and they shall follow me. For whoever wishes to save their life, will lose it; but whoever loses their life for my sake and the sake of the good news, will save it."
Jesus' call comes in three parts: deny, take up, follow. There is no taking up of the cross and following Jesus unless there is also, and firstly, self-denial. What does mean to deny the self? It is explained in the second verse: a losing of one's life for the sake of Christ and his good news. It means, then, that one must give up trying to save one's own life: the pursuit of every comfort and protection of this world in service and subjugation to the "I", the ego-driven self-centering that traces itself all the way back to Adam and Eve's transgression, their desire to be like God and place themselves at the centre of their own universe. In place of this, there is a placing of Christ at the centre of our own universe, so that all that we are and all that we have is in service and subjugation to him.
Luke 9:23-23 And he was saying to all, "If anyone wants to come after me, they shall deny themself and they shall take up their cross daily and they shall follow me. For whoever wishes to save their life, will lost it; but whoever loses their life for my sake, they will save it."
It is only Luke's version that adds "daily". One can only be crucified by the Romans once, but death to self and the imitation of Jesus precisely in this, must happen daily. And so we die to ourselves daily. At the same time, we die to the world. The world, with all its riches and comforts and promises and succour and the tempting offer of a full, self-fulfilling life without Christ, denying Christ, in the end we must die to it, precisely as it puts Jesus himself to death.
In this paradox we find life. Christ himself is the one put to death who rises to life, and our death is first an identification and union with Christ in his death. This first death is by our union with Christ by faith, and is signified in the death and resurrection symbolised by baptism. In his death we have died to self and died to the world, and come to life again. So that whatever death follows, and death in this life always comes to us, is already swallowed up in the sure and certain hope of resurrection.