dkuper's answer is a nice explanation of how mathematicians understand truth, the related notion of provability (the nature of this relationship is probably the motivating topic for the whole of mathematical logic), and the whole can of worms engendered by the notion of independence.
I want to address something else. You also gave an example of the sentence
This statement is not true.
for which things are a little different. As you noted, within a two-valued logic framework, this statement cannot be true or false!
The usual solution is to restrict the notion of mathematical truth to sentences written in a specific formal language. In fact, it is precisely to avoid these self-referential paradoxes that we spend all that time in elementary logic classes defining well formed formulas (or wffs).
If $\mathcal{W}$ is the set of well formed formulas, then a truth function is a function $v:\mathcal{W}\to\mathcal{V}$, where $\mathcal{V}$ is the set of allowable truth values. The set $\mathcal{V}$ is usually assumed to have some algebraic structure (a lattice usually), and $v$ is required to preserve that structure in some way (you can think of truth as a homomorphism!).
(The most common structures for $\mathcal{V}$ are the set $\{\top,\bot\}$ (two valued logic), some other Boolean algebra (classical logic), a Heyting algebra (intuitionistic logic), or a fuzzy lattice such as $[0,1]$ (fuzzy logic).)
By defining the set of grammatically permissible sentences recursively, such self referential sentences are hard to construct (although not impossible, as Godel showed), but more importantly, there is no way to introduce the truth function within the language - it is something that operates on the language from without.
(Provability is something else, which is why Godel was able to code the self referential sentence you cited before. The fact that unlike provability, truth can't be moved inside the framework in any reasonable way is where Tarski's undefinability of truth theorem, noted in dkuper's answer, comes in.)
So, the bottom line is that truth is a function, the domain of that function is carefully constrained, and the example of the sentence you gave above lies outside that domain. Its truth is therefore undefined, just as the reciprocal of $0$ and the square root of $-1$ are.