Consequentialism and Suffering
The standard consequentialist stance is that as act that reduces the amount of suffering is considered a morally good act, while one that increases suffering is a bad or evil act. Consequentialism seeks to minimize the amount of suffering.
Suffering is a Side Effect
It is not actually "suffering" that drives moral behavior. Suffering is more of a byproduct or side effect.
Hunger is the easiest example to explain: hunger (the feeling) is not itself a root level problem. Hunger is an affect that evolved to tell people when their energy supply is getting low. It is critical that the body never run completely out of free energy.
Imagine a tribe where half the people enjoyed feeling hungry, and the other half hated it. The ones that enjoyed it starved to death quickly. The only survivors are those that hated the feeling of hunger enough to do something about it, and therefor survive.
When we say it is immoral to starve a person, it is NOT because they suffer when hungry, but because the hunger is an indicator that the body is dangerously low on energy and might die. Morals are about life and death, not suffering.
Moral acts don't feel good because they are good. They feel good because we are the ancestors of people who felt like doing the right things.
Pain does not feel bad because it is morally bad to do things that cause pain, but the other way around: we are the ancestors of people who felt pain as being bad, and thereby avoided doing that action.
For example, poking a stick in your eye hurts. Imagine if it felt good. If poking your eye with a stick felt good, everyone would be doing it, and a lot of people would be blind. Blindness is a big disadvantage for survival. Thus anyone who felt pleasure at this died out long ago (long before humans, probably).
The reason a moral act is good, is because it helps you and your tribe to survive. It just happens that we evolved to suffer more when we do things that threaten our survival. The goal of morality is not to reduce suffering, but it happens to do this because survival and suffering are linked.