Great question, one that bothers a lot of people. Frankly, there's no easy answer. But there is an approach we've found helpful in trying to at least partially understand why God allows suffering and evil in the world.
First of all, we need to understand that it is people, not God, who cause most of the evil and suffering in the world. C.S. Lewis once said that four-fifths of the evil in the world is done by humans to other humans. Humans kill, torture, rape, force children into slavery, etc. Humans do this because the human race is sinful and in rebellion to God.
The other one-fifth of the evil is cause by nature--floods, tsunamis, earthquakes, famine, etc. Even in the natural world, sin is the culprit. The Bible says, "For the creation was subjected to futility" and "the whole creation has been groaning together in the pains of childbirth until now" (Romans 8:20, 22).
So why doesn't God do something about it? Why doesn't he stop humans from doing evil? Why doesn't he fix the world so natural disasters no longer happen? Well, we know that someday God is going to fix the world. He is going to deal with sin. But he is waiting for one very important reason: so that more people will turn to him. The Bible says, "The Lord is not slow to fulfill his promise toward you, not wishing that any should perish, but that all should reach repentance" (2 Peter 3:9).
Joni Eareckson Tada, has been a quadriplegic since she was paralyzed by a diving accident hen she was 18 years old (she's now 60 years old). Recently Joni was diagnosed with breast cancer. She has suffered terribly in her life, yet she has done amazing things throughout her life to help alleviate the suffering of others. Recently in an interview with Christianity Today, Joni answered the very same question you are asking: Why does God allow evil and suffering? Here's what she said:
"When people ask that question--even I struggle with that question--we aren't accepting the fact that this earth is wired to be difficult. The rule of thumb is that we experience much suffering because we live in a fallen world, and it is groaning under the weight of a heavy curse. If God being good means he has to get rid of sin, it means he would have to get rid of sinners. God is a God of great generosity and great mercy, so he is keeping the execution of suffering. He's not closing the curtain on suffering until there is more time to gather more people into the fold of Christ's fellowship."
We hope this helps a bit. The other thing you can do is to respond to God's call on your heart to do what you can to help relieve the suffering of others. Many people are feeling this nudge of God and are responding by supporting ministries and agencies that are working in places where the effects of both human and natural evil are the greatest. Many young people are actually going to these places to work and give of their time and talent. Maybe God is calling you to do something like that.