Enormously complex question. Hundreds of books have been written about it.
First, contrary to some of the other answers, we will assume that there IS a God and that He IS all-powerful (see Romans chapter 1). Being omnipotent, He COULD eliminate evil, but we observe that He doesn't. Therefore, He must have some use for evil.
We are conditioned by the experience of the human exercise of power to think of grand, often violent displays. Power can also be extremely subtle. If we also assume that God is all-knowing and wise, we can see (in a limited fashion) that His power and wisdom allow Him to manage evil so that it serves His purpose, even though it is not part of His ultimate goal. This is a God who uses even the anger of men to bring Himself praise (Psalm 76:10)
One of the purposes served by evil is to evoke exactly the reaction that you express. If we are at all morally aware, we should be horrified by it and this should lead us to seek ways to counter it and alleviate suffering. The presence of suffering should promote compassion. If we are motivated to seek an end to evil and suffering, we are motivated to seek God, who is their opposite.
Sometimes, when we are motivated by godly compassion, we are allowed to see examples of God's power applied directly. I have seen people healed, restored to their right minds, and even one raised from the dead.
When bad things happen to us personally, it may be the effect of our own participation in evil. In order for there to be anything like justice in the universe, there must be consequences for all actions, good and evil. They're just not always the consequences that simple, direct reasoning would lead us to expect.
This idea that there is such a thing as justice or fairness or equity is based on another assumption -- that there is a moral component to the universe as well as the physics and mechanics. That moral component must also come from God. Given this powerful, wise and just God, and given that we have all, at some point or another , offended against the moral structure of the universe ("all have sinned and fallen short of the glory [moral perfection] of God", Romans 3:23), the question is not, why do people suffer, but rather, why do good things happen to anyone?
Personal suffering may come to get our attention because God is trying to tell us something about Himself or about ourselves. My wife's cancer certainly got OUR attention. Or it may be that the 'bad' thing is merely necessary to direct us along a certain course (Check out the biographies / testimonies) of people like Elizabeth Eliot, Joni Eareckson Tada and David Ring). 'Bad' in this case, is a relative thing. Joni's quadraplegia and Ring's cerebral palsy are certainly not 'normal' or comfortable, but these 'handicaps' have given them the opportunity to act for God among people to whom they never could have related were they healthy. It is a mistake, born of middle-class values, to think that God is at all interested in maing us comfortable. The believer may rightly consider his own suffering to be part of the sufferings of Christ that lead to the redemption and restoration of creation.
The thing to realize here is that evil is not a mistake. It is not a flaw in God's plan. It is, if you will allow the Biblical analogy, the 'smelting fire' necessary to refine human nature and produce the type of person God is looking for.
The possibility of evil is the natural consequence of free will. Even God cannot FORCE someone to love Him -- obey Him, serve His lesser purposes, yes, but love cannot be forced without ceasing to be love. Love (and trust and faith) are choices. If we have the ability to choose to love, we also have the choice to reject. That rejection results in the moral consequences above. That the rejection is, at some point, universal (we have all rejected God at some times) means that suffering is also universal.