This is a question that has been discussed for thousands of years, and is still very relevant to those who want to understand the motivations of non-believers or who are questioning faith.
I'm absolutely not an expert about this, but I hope I can give a brief summary of how morality can be established without any supernatural basis.
The first place to start is evolution. There is a good book by the philosopher Patricia Churchland called "Braintrust - what neuroscience tells us about morality". It's an easy read and explains how our moral sense evolved over hundreds of millions of years. We have evolved as a parenting species and social species and that means we have developed empathic abilities that mean that we (or at least those of us who are healthy) share the feelings of others. We find joy in in the happiness of others and we experience pain when we see suffering. This is an objective part of our biology.
That suffering and happiness are objective biological facts gives us a basis for a naturalistic morality, given the reasonable understanding that suffering is to be avoided and happiness to be encouraged. We use a combination of discussion with others and our empathic senses to work out frameworks of morality that reduce suffering and encourage happiness. Of course, there is nothing simple about working out such frameworks, but the foundation of thought behind this is sound and based on objective facts - the existence of mental states of others, determined by discussion and empathy.
There is another way to look at how to act morally in the world without gods that can also be successful and that is the very practical guidelines given by the core of Buddhism: life involves suffering and we want to reduce or avoid suffering. One important aspect of avoiding suffering is to consider the consequences to others of your actions, because of Karma, which can be thought of as the reasonable view that you experience the consequences of your actions when they affect others. Example: if you are a jerk to others, you can expect them to be a jerk back to you! Buddhism suggests that to help live a happier life yourself it's a good idea to be kind and supportive of others. There is no notion of absolutes here, no concept of holiness or sin, just a practical recipe for a happier life.
Finally, part of living without the idea of supernatural moral absolutes is to trust yourself. You are an organism that has resulted from evolution over hundreds of millions of years to produce a successfully social species. Unless you have psychopathic tendencies you have a mental framework that will encourage you to do good and inhibit you from doing bad. Trust your neurons!