First-order logic is complete & sound. The notion of truth used here is model-theoretic.
Informally Godels incompleteness theorem says that for a sufficiently strong formal language there are truths that cannot be proved. How is truth encoded in the language? Is this actually correct, should it be that there are statements which cannot be either proved or disproved - that is they are undecidable. Does this mean that they actually have no truth status?
On
The incompleteness theorems (for a fixed theory) have as an assumption that the theory is consistent. This assumption can be phrased as a statement $C$ in the language of Peano arithmetic. Meanwhile, the first incompleteness theorem produces a Gödel sentence $G$ in the language of Peano arithmetic that is independent of the theory.
The Gödel sentence $G$ is true in exactly the same sense that the consistency assumption $C$ is true.
In other words, as soon as you work out a sense in which you think that $C$ is true, $G$ will also be true in that sense. There are many different "senses" that can be used here:
Semantically: $C$ is true in the standard model, and $G$ is also true in that model.
Disquotationally: $C$ is true, disquotationally, as a statement about natural numbers, and so is $G$.
Formally: If $C$ is provable in some reasonable metatheory, $G$ will also be provable in that metatheory. Here any metatheory at least as strong as PA is "reasonable". In fact PA can be replaced with a weaker metatheory such as PRA (primitive recursive arithmetic) or any other metatheory strong enough to prove the first incompleteness theorem.
"True" here means "true in the standard model." For example, the standard model of PA is the "actual" natural numbers. I agree that this is confusing. A more model-agnostic way to state the incompleteness theorem is that there are statements that are true in some models but false in others (hence that are neither provable nor disprovable).