Let them deny themselves and take up their cross daily and follow me. (Lk 9,23)
Following Jesus is not following any dogmas and rules and rituals. It is following a person. It is going behind a person; literally, it is following the foot steps of the leader. That is why when peter rebuked him saying that he should not go to Jerusalem; Jesus answered him saying that he should come after him. In fact he tells him “Get behind me Satan” because your thinking is not in line with that of God.
So a follower is to get behind the master and walk and the disciple should in no way take up the leader’s role in front of the master.
If the master is going to have an end, that is, going to a death on the cross the disciples also should be ready for that. Here the cross is not the daily difficulties and problem one faces. These are there even for the non-believers.
For Jesus the cross meant that he was alone, that he was insulted and humiliated because he followed the way of God. He wanted to do the will of the father. In the same way a disciple should be ready for a death that will included insults, humiliations, loneliness etc.
All these crosses come and they come daily because they follow Jesus. This is the self denial that Jesus speaks here.