To insure that there is no misunderstanding of what falls in each category, I have expanded on the prayer: Lord, grant me the serenity to accept the things I cannot change - which is the way my life is right now at this moment; the courage to change the things I can - which is how I relate to myself and others, the thoughts I think, the words I speak, and the actions I take; and the wisdom to know the difference.
As the Buddha taught when he first turned the wheel of the dharma, the cause of our suffering is our cravings. And cravings are a result of our not accepting things as they are. They seek to change our life, whether regarding personal maters, finance, status, whatever.
Now there is nothing wrong or unspiritual about wanting to change your life in ways that do no harm to others. But when that desire is based on a lack of acceptance, which is often quite adamant, for the way things are now, that desire becomes a craving. We become attached to it. We certainly do not approach it with equanimity. And thus we suffer.
Whereas if we work to change things in our life while accepting things are the way they are because it's just the way it is, we approach that work with equanimity, with non-attachment. If it happens, great; if it doesn't happen, that's ok too.
The other aspect of the serenity prayer gets to the point that for the most part we have no control whatsoever over whether what we strive for will happen; it usually depends on the actions of others or it may just be an impossible fantasy. We don't understand the distinction between what we have control over and what we don't. And so we suffer.
What we do have control over is how we approach life, how we live life. We thus have the power to create our own reality, which is a reality free of emotion in which we know we will be ok, safe spiritually, regardless what life throws our way because we have returned home, and will always return home, to our true Buddha self and so be at peace and happy. We have the power to create nirvana for ourselves (see my posts, "Nirvana - It's Right Before Your Eyes," and "You Can Be in Control").
The reality which we typically experience, however, has been created unconsciously by our ego-mind and its emotions, judgments, cravings, and attachments. And so we live in a state of endless frustration and samsara.
The "trick" with the serenity prayer, or indeed with any efforts to make real progress on the spiritual path - like following the Noble Eightfold Path, is that until one frees oneself, or at least begins to free oneself, from the control of your ego-mind, it will not allow you to do Right anything or accept anything.
Thus, if you want to end your suffering, the focus of your spiritual work should be on freeing yourself from the control of your ego-mind. That is what will open up dharma gates for you. While freeing yourself from the control of the ego-mind is no easy matter, the path to do that is clear (see my post, "How to Free Yourself from the Control of Your Ego-Mind").
May you experience peace and happiness.