How are we to react to our suffering? The answer is unique to Christianity.

We are members of the Royal Priesthood, together as one in the Mystical Body of Christ
Just as in the Old Testament, Israel of the New Covenant is made of priests:

I Peter 2:9-10
But you are a chosen generation, a kingly priesthood, a holy nation, a purchased people: that you may declare his virtues, who hath called you out of darkness into his marvelous light: Who in times past were not a people: but are now the people of God. Who had not obtained mercy: but now have obtained mercy.
Our being priests means that we make sacrifices, we offer something. The ordained Catholic priest offers, as a representative of Christ, Sacrifices at the Altar for those who say "yes" to Christ's invitation to share the fruits of Calvary, just as the ministerial priests in the Old Testament offered sacrifices for the sins of the people. But what do we of the non-ministerial royal priesthood offer? We offer ourselves -- our bodies, hearts, praise, gratitude, worship, joys, works, and our sufferings.

Why do we do this? Because we are exhorted to "put on Christ" and to imitate Him, our High Priest and Spotless Victim, so that we might partake of the divine nature. In order to redeem us, Our Lord took on flesh and gave all to the Father; in order to be Christ-like, we, too, must take up our cross, accept suffering, and strive to offer Him all:

Luke 14: 27
And whosoever doth not carry his cross and come after me cannot be my disciple.

II Corinthians 4:8-12
In all things we suffer tribulation: but are not distressed. We are straitened: but are not destitute. We suffer persecution: but are not forsaken. We are cast down: but we perish not. Always bearing about in our body the mortification of Jesus, that the life also of Jesus may be made manifest in our bodies. For we who live are always delivered unto death for Jesus' sake: that the life also of Jesus may be made manifest in our mortal flesh. So then death worketh in us: but life in you.

Galatians 6: 14
But God forbid that I should glory, save in the cross of our Lord Jesus Christ: by whom the world is crucified to me, and I to the world.

Philippians 3:8-11
Furthermore, I count all things to be but loss for the excellent knowledge of Jesus Christ, my Lord: for whom I have suffered the loss of all things and count them but as dung, that I may gain Christ. And may be found in him, not having my justice, which is of the law, but that which is of the faith of Christ Jesus, which is of God: justice in faith. That I may know him and the power of his resurrection and the fellowship of his sufferings: being made conformable to his death, If by any means I may attain to the resurrection which is from the dead.

I Peter 2:19-22
For this is thankworthy: if, for conscience towards God, a man endure sorrows, suffering wrongfully. For what glory is it, if, committing sin and being buffeted for it, you endure? But if doing well you suffer patiently: this is thankworthy before God. For unto this are you called: because Christ also suffered for us, leaving you an example that you should follow his steps.
Think of Christ in the Garden, under so much stress and agony that He literally sweated Blood. Think of Him being hounded and mocked by people who should've known Who He was. Think of Him with a crown of thorns thrust onto His head, being spat upon, beaten, and nailed to a Cross. God Himself suffered in His human nature; why should we be spared? And now think of Him in Heaven, pouring out onto us the graces of His once and for all Sacrifice at Calvary during the unbloody re-presentation of that Sacrifice during the Mass. He is perfect, He suffered (His Sacred Heart is still wounded by our sins!), and He offers Himself yet up to the Father -- and to us for our redemption. We are called to offer ourselves up to the Father and for others, too.

Our imitation of Him and our gifts to Him, though they are nothing without His Sacrifice, build up the Body of Christ if they are joined to His sufferings:

I Corinthians 12:26
And if one member suffer any thing, all the members suffer with it: or if one member glory, all the members rejoice with it.

Colossians 1:23b-24
...whereof I Paul am made a minister. Who now rejoice in my sufferings for you and fill up those things that are wanting of the sufferings of Christ, in my flesh, for His body, which is the church:
Now, is Paul saying that Christ's sufferings and Sacrifice weren't enough? Is he "taking away from Christ" by saying that we are to "fill up" those things that are "wanting" in His sufferings?
No, of course not.
He is saying, though, that we are One Body, that we co-operate with God in profound ways ( I Corinthians 3:9 "For we are God's coadjutors [co-workers, assistants]..."), and that, in an inscrutable way, our sufferings benefit one another.