Is good the absence of evil or is evil the absence of good?What do religions tell us?
In the neo-Platonic tradition, from Plato, through Plotinus, and from there integrated into mainstream Christian and Islamic theology, good is the only true reality, and evil is just the absence of good, just as darkness is the absence of light, and cold is the absence of heat.
But there are other religious and philosophical traditions in which goodness is taken to be the default state, and evil is a real but negative imposition on top of it, in which case you could arguably call good the absence of evil. There are also dualist traditions, such as Zoroastrianism and Manichaeism (which continues to influence some variants of Christianity) in which good and evil are both real but are equal (or nearly equal) and opposite forces.
The question of how to behave alters based on your assumptions. In a neo-Platonic world, you focus on the good, because the evil is illusionary, whereas in the Zoroastrian tradition, evil becomes something that must be actively driven out. It should also be mentioned that there are traditions, such as Taoism, in which neither good nor evil is an important concept (and which is instead focused on "balance" and "imbalance" between opposing, but not morally opposed forces), as well as others in which both good and evil are seen as being largely illusionary (a concept which also informs Nietzsche's book Beyond Good and Evil).
(Not really an answer but it's something that is missing in the question but could be important.)
I want to add that there seems to be a major difference between evil and morally wrong, on top of the relation between - using different terms here for clarity - moral goodness and moral wrongness that Chris describes in his answer. The SEP has a decent overview which aspects might play into a concept of evil. The historical overview there is a bit lacking though, so especially some more knowledge on Kant's and Arendt's ideas on evil might be good to know.
In short, one idea is that we might differentiate between evil acts and only morally wrong acts. For example, evil acts could just be acts which are very morally wrong. Or evil acts might need some sort of evil intention or motivation behind them. When someone does a morally very wrong act while thinking it to be a right act then the intention is different from a non-deluded sadistic serial killer.
Another issue lies in the relation between evil acts and evil character. We could try to argue the easy route by claiming that evil people are those which do many evil acts. But perhaps we prioritize the character over the acts and argue that evil acts are those which bring a character to become evil or only are expression of it. Furthermore it's possible that we're using the word evil for different concepts when it comes to acts compared to character.
So why is it important to think about the concept of evil? We already use the term evil in order to point out something especially morally blameworthy. When we call someone evil we basically make him seem less human. But this might not always be true as, like mentioned, merely doing very morally wrong acts might not be enough to get a different status in blameworthiness and monstrosity.