There are many forms of suffering, such as, disease, disability, natural disasters, war and conflict, or persecution.
Probably the most well known passage about suffering is Philippians 1:12-14:
"12 Now I want you to know, brothers and sisters, that what has happened to me has actually served to advance the gospel. 13 As a result, it has become clear throughout the whole palace guard and to everyone else that I am in chains for Christ. 14 And because of my chains, most of the brothers and sisters have become confident in the Lord and dare all the more to proclaim the gospel without fear."In this passage we see that Paul explains that his suffering, prison at this point, actually serves to advance the gospel (v. 12). In fact, it was well known that Paul was in prison because of his faith in Christ (v. 13), everybody from the palace guard to the commoners knew.
Due to the fact that everyone knew this, other believers who heard about Paul's imprisonment, or suffering, "have become confident in the Lord and dare all the more to proclaim the gospel without fear" (v. 14). If Paul would have believed that prison was the end, then he would have missed a golden opportunity to share the gospel.
That's not the only time Paul writes about suffering:
"3 Not only so, but we also glory in our sufferings, because we know that suffering produces perseverance; 4 perseverance, character; and character, hope. 5 And hope does not put us to shame, because God’s love has been poured out into our hearts through the Holy Spirit, who has been given to us." (Romans 5:3-5)
"For our light and momentary troubles are achieving for us an eternal glory that far outweighs them all." (2 Corinthians 4:17)
I know that I haven't explained why suffering goes on, it's a mystery that mankind will never solve. God doesn't want to hurt his children, but He does use suffering for his purposes. Whatever Satan dishes out for evil, God turns it around and uses it for good. Even when Satan had the plan to kill Jesus Christ, God showed that Jesus' body could be tortured and destroyed, but He will never die. Jesus, instead of dying, conquered death to become our salvation, our Savior.
Suffering is not a good thing, let me state the clearly. Suffering is terrible, however, God can still use it for good, as we see with the apostle Paul. In times of suffering God will give strength, patience, perseverance, peace, or whatever is needed to those who are suffering who will ask Him and He will use that evil for good.
I'll end by sharing one of my personal favorite Bible verses, Romans 8:28:
"And we know that in all things God works for the good of those who love him, who have been called according to his purpose."