Moral Values do not Require Gods or Religion:
A popular claim among religious theists is that atheists have no basis for morality that religion and gods are needed for moral values. Usually they mean their religion and god, but sometimes they seem willing to accept any religion and any god. The truth is that neither religions nor gods are necessary for morality, ethics, or values. They can exist in a godless, secular context just fine, as demonstrated by all the godless atheists who lead moral lives every day.
Love & Good Will:
Good will towards others is vital to morality for two reasons. First, genuinely moral acts must include a desire that others do well it's not morality to grudgingly help someone you wish would curl up and die. It's also not morality to help someone due to inducements like threats or rewards. Second, an attitude of good will can encourage moral behavior without needing to be prodded and pushed. Good will thus functions as both a context and driving force behind moral behavior.
Some may not immediately recognize the importance of reason for morality, but it's arguably indispensable. Unless morality is simply obedience to memorized rules or flipping a coin, we have to be able to think clearly and coherently about our moral choices. We have to adequately reason our way through the various options and consequences in order to arrive at any decent conclusion. Without reason, then, we cannot hope to have a moral system or to behave morally.
Compassion & Empathy:
Most people realize that empathy plays an important role when it comes to morality, but just how important it is may not be as well understood as it should be. Treating others with dignity does not require orders from any gods, but it does require that we be able to conceptualize how our actions affect others. This, in turn, requires an ability to empathize with others an ability to be able to imagine what it's like to be them, even if only briefly.
Without personal autonomy, morality is not possible. If we are simply robots following orders, then our actions can only be described as obedient or disobedient; mere obedience, however, cannot be morality. We need the ability to choose what to do and to choose the moral action. Autonomy is also important because we are not treating others morally if we prevent them from enjoying the same level of autonomy which we require for ourselves.
In Western religions, at least, pleasure and morality are often diametrically opposed. This opposition is not necessary in secular, godless morality on the contrary, seeking to generally increase the ability of people to experience pleasure is often important in godless morality. This is because without any belief in an afterlife, it follows that this life is all we have and so we must make the most of it while we can. If we can't enjoy being alive, what's the point of living?
Justice & Mercy:
Justice means ensuring that people receive what they deserve that a criminal receives the appropriate punishment, for example. Mercy is a countervailing principle which promotes being less harsh than one is entitled to be. Balancing the two is key for dealing with people morally. A lack of justice is wrong, but a lack of mercy can be just as wrong. None of this requires any gods for guidance; on the contrary, it's common for stories of gods to depict them as failing to find balance here.
Honesty is important because truth is important; truth is important because an inaccurate picture of reality cannot reliably help us to survive and understand. We need accurate information about what is going on and a reliable method for evaluating that information if we are to achieve anything. False information will hinder or ruin us. There can be no morality without honesty, but there can be honesty without gods. If there are no gods, then dismissing them is the only honest thing to do.
Some deny that altruism even exists, but whatever label we give it, the act of sacrificing something for the sake of others is common to all cultures and all social species. You don't need gods or religion to tell you that if you value others, sometimes what they need must take precedence over what you need (or just think you need). A society without self-sacrifice would be a society without love, justice, mercy, empathy, or compassion.
Moral Values Without Gods or Religion:
I can almost hear religious believers asking "What's the basis for being moral in the first place? What reason is there to care about behaving morally at all?" Some believers imagine themselves clever for asking this, certain that it cannot be answered. It's only the cleverness of a teenage solipsist who thinks he has stumbled on a way to refute every argument or belief by adopting extreme skepticism.
The problem with this question is that it presumes that morality is something that can be separated from human society and consciousness and independently grounded, justified, or explained. It's like removing a person's liver and demanding an explanation for why it and it alone exists while ignoring the body they've left bleeding out on the ground.
Morality is as integral to human society as a person's major organs are integral to the human body: although the functions of each can be discussed independently, explanations for each can only occur in the context of the entire system. Religious believers who see morality exclusively in terms of their god and religion are as unable to recognize this as someone who imagines that humans acquire a liver through a process other than through the natural growth that lies behind every other organ.
So how do we answer the above question in the context of human society? First, there are two questions here: why behave morally in some particular set of circumstances, and why behave morally in general, even if not in every case? Second, religious morality which is ultimately based on the commands of a god cannot answer these questions because "God says so" and "You'll go to hell otherwise" don't work.
There is insufficient space here for a detailed discussion, but the simplest explanation for morality in human society is the fact that human social groups need predictable rules and behavior to function. As social animals, we can no more exist without morality than we can without our livers. Everything else is just details.