I'll focus on
I had always assumed there was a good and evil while I was a
Christian, but now being an agnostic and delving into philosophy, I
don't know if this is a correct assumption. Does anyone know a solid
proof for the existence of a moral right (good) and a moral wrong
(evil)? I may be asking an impossible question, but I think the burden
of proof lies on the theist to prove morality even exists
User idiot an has already sort of given a brief overlook, but I want to make a few points hopefully more precise.
Firstly, the burden of proof isn't strictly a concept in philosophy. As a rule of thumb: whenever someone makes a claim that needs backing then an argument has to be made. When there's disagreement then there's no default position that can be asserted without argument. So moral anti-realism / nihilism / scepticism isn't a default, but would also have to be "proven" itself by providing an argumentative challenge.
Secondly: we should ignore the question of whether good and right, or bad and evil are the same things. This is another matter, but we're asking about whether there's morally right or wrong. For example we might think that something can be morally wrong but not necessarily evil, as evil seems to imply intention. But this doesn't have to be the case. So we bracket this issue.
Thirdly, there's the issue of terminology: what are we looking for when we want to find out whether morally right and wrong exist? This is a question of metaethics. But metaethics has many different aspects and disagreement between which aspects are even relevant in discussion. So we can ask semantically: how do moral propositions work and what are they refering to? Or metaphysically/epistemological: how can moral facts be known or defended? And so on.
"Moral realism" is mostly - but not only - a description of a semantic position. It at least holds that moral statements have a truth value, which can be true in some cases and refer to mind-independent facts. But this wouldn't work without it being possible to know moral facts, so this already plays into it. Also it's not clear what constitutes mind-independent facts.
Usually moral realism gets divided into non-naturalism and naturalism. Naturalism holds that moral facts derive from natural facts. But there are also subjectivist naturalist positions or positions. The difference depends heavily on the specific position. Non-naturalism holds that moral facts are abstract. Here the math example comes in handy as an analogy: mathematical facts seem to be abstract, moral facts might be thought of to be similar. At least non-naturalism thinks they can't be derived from natural facts alone.
To show an example of why the positions aren't clear. Some forms of moral constructivism (more on that here) hold that people have basic values. Those might or might not be similar; some hold that we as humans have the same basic values. Now any moral beliefs or acts that contradict those basic values are incorrect for us. Hence, in a way, we might think that there's objectively right and wrong. On the other hand it looks like an anti-realist position. But Scanlon for example considers himself to be moral realist, while Street thinks her position is sort of relativist.
Tl;dr: moral anti-realists or relativists in the usual sense make argumentative challenges for position that are "effectively realist", in that they permit judging other peoples moral belief to be right or wrong. There's an enourmous amount of literature, so ideally it would be best not to assume anything. From what is seems most professional philsophers are atheist and hold some sort of moral realism. But this does not mean there aren't defensible moral anti-realist or relativist accounts, but instead that moral realism shouldn't be dismissed without reading the arguments.