I think you have to look at it by different schools of thought:
1) Ayn Rand is a well known for believing in an objective moral truth while not believing in a higher power. From the Wikipedia article:
Objectivism's central tenets are that reality exists independent of consciousness, that human beings have direct contact with reality through sense perception, that one can attain objective knowledge from perception through the process of concept formation and inductive logic, that the proper moral purpose of one's life is the pursuit of one's own happiness (rational self-interest), that the only social system consistent with this morality is one that displays full respect for individual rights embodied in laissez-faire capitalism, and that the role of art in human life is to transform humans' metaphysical ideas by selective reproduction of reality into a physical form—a work of art—that one can comprehend and to which one can respond emotionally.
In such a world view, it is not obvious that "to help out others to one degree or another" is necessarily the moral thing to do.
2) Closely related would be a morality based on the two works of Adam Smith, The Wealth of Nations and The Theory of Moral Sentiments. Together, they argue for a laizzes-faire that acknowledges what many observe: that we generally want to see others be better off:
How selfish soever man may be supposed, there are evidently some principles in his nature, which interest him in the fortunes of others, and render their happiness necessary to him, though he derives nothing from it, except the pleasure of seeing it. Of this kind is pity or compassion, the emotion we feel for the misery of others, when we either see it, or are made to conceive it in a very lively manner. That we often derive sorrow from the sorrows of others, is a matter of fact too obvious to require any instances to prove it; for this sentiment, like all the other original passions of human nature, is by no means confined to the virtuous or the humane, though they perhaps may feel it with the most exquisite sensibility. The greatest ruffian, the most hardened violator of the laws of society, is not altogether without it.
3) A utilitarian might come to a similar conclusion as you (that it is generally good to be nice to others), but it would depend on an individual's perceived utility curves. For example, depending on those curves, a utilitarian may come to the conclusion: "Do unto others as others would do unto you up until this point".
4) Marx is an obvious example of a secular morality. In that world view, it's not so much "Do unto others..." as it is looking at what sort of things are owed to different people based on their class (labor is due certain things, current owners of capital are due their come-uppance, etc.)
5) I haven't read too much Richard Dawkins, but from what I have read he seems to present a world view that is different in some ways but related to the above. He believes in a very empirically based view on everything. Something exists if (1) you observe it with your senses, (2) you observe it through something that heightens your senses (such as a telescope or microscope), or (3) you can create a model for it based on (1) or (2). From that as a starting principle, you can deduce your world, including moral principles.
6) Descartes is a potentially interesting example as well. In his statement "I think, therefore I am," he does not take a higher being as his initial principle, but rather he takes his own cognition as the starting principle and derives God. This makes it, in some sense, a "secular" morality, in that traditional theological moralities (such as that of Augustine, Aquinas, etc.) take a divine power as the starting point, not the person.
So, I think these are few different schools of thought of a secular basis for moral reasoning.