Suffering
Eliminating suffering has the biggest impact on utilitarian happiness of any other change you can make.
"If there is a meaning in life at all, then there must be a meaning in suffering. Suffering is an ineradicable part of life, even as fate and death. Without suffering and death, human life cannot be complete." -Viktor E. Frankl
Due to habituation and the hedonic treadmill, you can only get marginally happier for some time before you go back to your baseline. However, there can be no end to suffering for those who are starving, lack shelter, have debilitating illnesses with chronic pain, or unexpectedly find themselves the target of a Spanish Inquisition.
Suffering in Religion
In many religions, but especially Buddhism, the focus is more on the elimination of suffering rather than the increasing of happiness. This reflects the nature of life at the time when these religions were created. There are many reasons why ancient life had so much more suffering, but the fact that the average person could expect to see the deaths of multiple children during their lifetimes should be sufficient evidence.
Modern life still has suffering, and reducing it is the fastest way to increase utilitarian happiness. However, focusing exclusively on eliminating suffering is insufficient when the average person experiences relatively little trauma, has free time, and disposable income. It can also lead to unnecessary rumination, since focusing attention on suffering can intensify it. Looking back at your life from the perspective of suffering can sometimes frame events in a negative context, when a positive perspective could be equally valid. Our thinking and attachment to suffering also causes suffering (it's all so meta).
"A problem cannot cause suffering. It is our thinking and attachment to it that causes suffering." -Buddha
The Christian manifestation of suffering focuses on the wretchedness, sinfulness, guilt and shame of the believer. While the intent is to be welcoming to all, even those who are at their lowest point in life, the message often fails to connect with those who don't see themselves as wretched. Most modern people have pretty good lives and, while we all have highs and lows, many of us can go through life without ever hitting "rock bottom." Helping those at their lowest is the most important thing, but the vast majority people live in the "spongy middle" and just need help getting to the top.
The Eightfold Path is the Buddhist practice designed to eliminate suffering in the mind through mindfulness practice, mixed with some Ten Commandments style advice. A significant amount of suffering is self-imposed by the mind, and mindfulness practices can help eliminate rumination, desire, and other thought-loops that exacerbate negative emotions.
Suffering in Music
Lots of good music, poetry, comedy, and all other art forms are borne from suffering. This doesn't mean that suffering is good, just that it is an inevitable part of life so you should find some way to profit from your trauma.