As for moral relativism, the logical conclusion of moral relativism is moral nihilism. In a relativist ethical system, what is there to stop someone from making up his or her own completely new set of morals in which stealing and hurting innocent people and the like are all good? If morality is based on opinion, as moral relativists say, that person's perverse moral system is just as right as anyone else's.
Slightly off-topic, but - not quite. You are greatly oversimplifying the matter. "Your moral system is just as right as mine" is, in itself, a kind of moral judgment. The conclusion of your argument, which is framed as an absolute judgment of value, is in direct contradiction with its premise, which is that all judgments of value are relative.
If morality is relative, nothing stops me from claiming your moral system is perverse and wrong. It may not be *absolutely* perverse and wrong, but that is no reason for me to accept it. Morality is something people care a great deal about - you can make your own bizarre moral system, but then you will attract the ire of everybody who disagrees, which very well might be nearly everyone else. Moral relativism does not say that these people should logically view your system as equivalent, because that would create a moral absolute. In fact, it is quite the opposite: these people will consider that your morals are wrong and they will attempt to convince you to adopt their morals instead.
Morality is a sort of tug-of-war where everybody passionately tries to pull moral defaults towards their own standards. It is competitive, and in this context, the idea that morality is absolute is actually adaptive. It is easier/better for the most part to accept that idea, even though it is wrong. Moral systems are more convincing when they are more logically consistent, but they all need to be based on a core of disputable principles.
In the case of animals, you can easily justify eating them from the principle that an entity is only required to care about others in proportion of genetic closeness, or you can condemn it from the principle that all suffering must be avoided. There is no natural way to compare core moral principles and arguments for or against them are usually emotionally grounded (which is a fair tactic in this case). Logic cannot tell you what you want to do, it can only tell you how to do it, or whether your wants are consistent with each other. It is easy to compare our current morality with ancient morality and believe that we could convince our ancestors of the errors of their ways, and you might be correct, but there is a possibility that you would be caught off guard by very high logical consistency and sound arguments based on principles they take as self-evident but which you reject (just like we might all be vegetarian in the future and not see that there are legitimate arguments for eating meat - they are just based on different core principles). You are well within your rights to condemn slavery from a modern perspective (after all, we pretty much all agree *now*), but wishing for absolute morality does not make it exist.