If there is no objective morality, how could we determine ethics/morals. What is moral/ethical today may not be considered moral tomorrow. In future, even pederasty, pedophilia, bestiality may be considered normal. Some European countries have already legalized bestiality. Greek cultures have accepted pederasty, hedonism in the past. What is your opinion.
I'm certain I've answered this question before on AllExperts, but as the search function is currently glitchy, I can't find it to point you to it. So here's my quick reply.
There are, when it comes right down to it, three types of perceived morality:
1. Universal. This is a morality from an outside source that pertains to everyone. Morality from a deity, in other words. This type of morality, it is important to note, is generally inflexible. With minor exceptions, the "laws" of a religion don't change over time. For this type of morality to "work," every last person would have to agree 100% with that set of morals. And obviously, that's no more likely than having everyone agree on a single religion. And even within that religion, there will be believers who don't accept 100% of that faith's tenets.
2. Societal. This is, obviously, morality as imposed by one's society. The laws, generally. Laws, of course, can and do change over time, but not always in a way that makes sense, or as rapidly as they should. As you point out, this cannot apply to everyone because the "rules" about what is acceptable vary from one nation to another. Or even within that society - we have things that are 100% legal in my society that I consider to be utterly unethical.
3. Personal. In other words, this is a moral code of an individual person as developed over the course of a lifetime. And it applies only to that individual person. And these morals change over time, as we (hopefully) gain more wisdom as we age and have a better understanding of the world.
So the idea of an "objective" morality, as you put it, is simply impossible. Everyone is different. Societies are different. Religions are different. Of the three types listed above, the only one that can truly work is the personal morality. Of course, I mean that subjectively. My morality probably wouldn't "work" for you, and yours probably wouldn't for me.
But aside from all that, I think it's not particularly difficult to determine what is or is not moral. Human beings are tribal creatures. To my mind, anything that is harmful to the tribe is "immoral." So the obvious things like murder, theft, etc. are going to be "universal," as far as that's possible. It doesn't take an outside source of morality to make this obvious. If you kill another member of your tribe, for example, you have fewer individuals to gather food or protect the tribe from predators. It just makes logical sense.
Obviously, these answers are very basic. Hundreds of others have gone into great detail on this question, so if you need to learn more, it's certainly out there. A simple Google search will point you to any number of sources.
Hope this is helpful.