Take up your cross daily in this context means to be ready to die under persecution for the sake of your faith in Jesus Christ. It does not mean take up a particular doctrine or teaching daily. The context is clear and simple.
If it is not done by Grace through Faith (in Christ and Him Crucified) it is law. No other way can you please and live for God that allows the Holy Spirit to work in you both to will and to do for His good pleasure. (Rom. 8:2 KJV, Phil. 2:13 KJV, Gal. 2:18, 21 KJV, Gal. 5:4 KJV, 1 Cor. 1:17-31 KJV)
James 4:10 says - Humble yourselves in the sight of the Lord, and he shall lift you up.
A humble Believer trust and Faith is placed and maintained exclusively in Christ and His Finished Work at Calvary Cross where the victory was won. This Believer is denying themselves by taking up their cross daily.
Luke 9:23 - And He said to them all, If any man will come after Me, let him deny himself, and take up his cross daily, and follow Me.
Denial of self is recognizing my moral and spiritual poverty; for I know that in me (that is, in my flesh) dwells no good thing (Romans 7:18). It is knowing and understanding that I am unable to serve God in my own will power, self-will, strength, ability, and power. We see this in the first three "Beatitudes."
Blessed are the poor in spirit: for theirs is the kingdom of heaven. Matthew 5:3
Blessed are they that mourn: for they shall be comforted. Matthew 5:4
Blessed are the meek: for they shall inherit the earth. Matthew 5:5
Take up his cross is seen in Matthew 5:6; Blessed are they which do hunger and thirst after righteousness: for they shall be filled.
I am filled by God's provisions (GRACE) found in Christ by placing my Faith exclusively in Christ and Him Crucified (Believing who Jesus is, and what Jesus has done). Therefore I receive all the benefits of this "Resurrection Life" by understanding that we have been planted together [Christ and ourselves] in the likeness of His Death [Romans 6:5, Psalm 103:2].