This is a question I have asked myself for quite some time. Recently I've seen a feature about scientists who claim morals have nothing to do with our religion, but with our upbringing and genes. What a drawback for the conservatives and fundamentalists (who won't believe this anyway, as it's science).
So what do those scientists really tell us?
Experiments show, they claim, that little children (about 2-4 years old, which means they've not received a 'moral education', or at least not a lot of it) show helpfulness as well as adults (or not...). This suggests that the will to help others of our kind is somehow build into our genes - just like instinctive behaviour of every creature on earth.
Other experiments, which were conducted with adult people, also show that we share more liberally than economists would give us credit for. When faced with the task of sharing a certain amount of money with another, unknown person, most prefer to offer the other one 50% (they do not share money they've earned, but money freely given by a third source). Economists would expect the offers to be something like 70-80% to the person itself and 30-20% to the one they're supposed to share with. Mostly Economists are wrong. Now, the interesting part is that the people taking part in this experiment came from different countries and different religions. Some were very religious while others weren't religious at all.
This shows that religion as a such (no matter which one) is not an important factor in our morals.
It is indirectly one, though, because society is something which defines our morals and society itself is to a certain degree influenced by the main religion.
This doesn't mean the basic morals differ a lot because of religion, since basically all religion damn the same sins and approve of the same actions. Hurting others and stealing from them is a sin, thus it is damned. Being kind to others and giving is approved of and thus a 'good deed'. Realizing this made me suspect that maybe our religions also come from our genes - if we tend to behave in a certain way towards others of our kind, is it unrealistic to think that our 'natural' behaviour will have an influence on our philosophies and religions? We may have stopped to give our gods a human body, as the people of Ancient times did. We may have stopped to see our gods as someone you can simply meet (even though at least the Christians somehow don't manage to keep away from seeing God as some sort of old man with a long, white beard). But we still see the morals as something that should go with what we feel is right.
This is the main difference between our moral feelings and the law. The law is based on our morals, but it doesn't change as fast as they may. So while something is still legal or illegal by the law book, this might have changed in our minds already.
But back to the question about the origins of morals.
If we inherit the most basic morals we have, then there would be no need to teach them to anyone - but we still don't know what the most basic morals are.
Most of our morals seem to come from the society we live in, though. We may be naturally inclined to help our own kind, but apart from that we are sometimes worse for our own species than any other creature. (Homo hominis lupus, just for those of you who had Latin at school, though I personally learned this saying from a novel for adolescents when I was about 12.) The way we act towards others is mostly determined by what the society approves of. As we all are social beings, we want to be accepted and respected by the society we live in (i.e. all other humans around us) and therefore we are ready to accept the rules of society (not to be confused with laws, see above) and at least openly agree with it's morals. What we think about them stays in our own mind.
So in essence moral isn't given to us by God - and if He approves of it or not we'll all learn after we've died, I think. We (as a society) make our own morals - and this is something especially conservatives should realize soon.