As you rightly point out, that there is no black and white objective standard that non believers look to when thinking about morals, ethics, or laws. There are a handful of sources that most non believers will pull from. And they likely pull from those different sources in different quantities or in different ways resulting in variation of what non believers think. I would also like to mention that my personal views of right and wrong and my views of what should be law are very different and informed differently.
Before I dig into what informs my views, I want to say that I try not to present my views as 'correct' or 'better' than anyone else's. Obviously, they appeal to me more, but, I am open to being wrong and I'm open to changing my mind. Being able to change my mind on a topic is a feature, not a bug. So, here is what I think informs my views (not necessarily in order):
1. Culture / Society. I think this is one of the obvious ones and has already been brought up by some of the posters above. And I think it applies to the religious as well, which is why an 18th century American Christian and a 21st century American Christian may not agree on everything. Even when we have a static and objective standard such as the text of the Bible, there are variations in what people believe and I attribute a lot of that to societal influence.
The culture and society that I've been brought up in has been massively influenced by Christianity and that absolutely colors my views. Concepts in Christianity that promote human dignity, moral responsibility to others, kindness, mercy, and charity are all ideas that I still value. While I reject the idea that Christianity 'owns' these values or that they are entirely unique to Christianity, I appreciate that some of these ideas, promoted by Christianity, are foundational elements in our society.
A question that has been presented to believers on this board in the past is: If you had, by chance, been born into a Muslim household in Afghanistan in a society that is 99.7% Muslim, what is the likelihood that you would be Muslim today and that you would share the values of that society? Its hard to answer that question, but I think most of us can recognize that the circumstances of where and when we are born absolutely matters. The uncomfortable suggestion here is that so much of what we all end up believing is almost accidental.
American society also definitely includes measures of skepticism of power and individualism and personal liberty. Americans, moreso than many other places in the world, have inherited a very strong sense of 'you can't tell me what to do!' type individualism that permits someone like me to question established norms without being excluded from that society. There is no shortage of times and places where any of us would be killed or persecuted for believing the 'wrong' thing.
All of this is important because I think it helps explain a current rise in non-believers in Western societies. For better or worse, when there is less pressure to conform to a single monolithic set of ideas, people are more free to choose ideas that they gravitate toward.
2. Empathy. With few exceptions, we all have a sense of empathy and ability to understand and care about the feelings of others. It informs an innate responsibility to care about others facing harm, hardship, or injustice. For example, while I may not share your views and your values, I can imagine an action taken to/against you and I can understand why you might feel either helped or harmed by that action. And if I respect you and your personhood, this informs how I might treat you. That is to say I can treat you how you want to be treated and not how I think you 'ought' to be treated according to my values.
I think that empathy, as I describe it, requires humility. Recognizing and respecting your perspectives requires me to consider a viewpoint other than my own. And it requires that I not assume that my views are perfect.
3. Human Well Being and Suffering. There is definitely a limit to which I place value in utilitarianism on this topic, but ideas of promoting well-being and minimalizing suffering definitely drives my views.
4. Personal Experience. Around the time that I left the Catholic church 24-ish years ago, I was dating someone that was Jewish and had a good friend that had come out as gay. It just happened that these two people were about the most decent and honest and kind and charitable people that I knew. I experienced some pressure from friends and a family member to be concerned with converting both of these friends in the interest of their salvation. Pressure from people that couldn't hold a candle to the decency I saw in these condemned friends. . . . and so my experience is that wonderful and beautiful people come in all many different philosophical shapes and sizes. There is an obsession from some that we must believe the 'correct' thing and pray the right way in order to be good. If God is to judge us all when we die for what is 'in our hearts', then my experience suggests to me that what God you pray to may have very little to do with who gets saved.
5. Others. Reason and Critical Thinking, Biology, Secular philosophy, virtue ethics, and probably a whole host of other things.
In response to your example of looting and theft being bad. I think we can simultaneously default to that simple baseline of 'stealing is bad' while also trying to understand a more complex social dynamic - if one exists. Someone looting an electronics store and stealing TVs to sell out of their van is not very defensible. The proverbial parent stealing a loaf of bread to feed their children is more complicated. The action of the theft is still wrong, but consideration of the motivation or situation of the person stealing can inform our reaction. As my wife would say, understanding why someone does something is about 'understanding, not justification.' If people need to steal just to be able to feed their children, should we as a society be concerned? Do we have a moral responsibility to them?
Determining right and wrong is difficult. Even when you believe in an absolute and objective source for all of morality, there is disagreement on what is and is not moral. If there is an absolute moral standard, it still seems reasonable to me that we should all admit that we are not that absolute moral standard. And if we believe in that absolute standard, we are still trying to understand it through biased glasses and we are all influenced by our cultures, our experiences, our feelings, and our reasonings.
The absence of God in the equation is scary. The absence of an objective standard is scary. The idea that rape, murder, and anything else can't be taken as wrong by some objective and cosmic source is scary. On the other hand, it frees up to consider moral questions for ourselves. Things can be allowed or disallowed because we as a society talked it over and decided.
