When we talk about validity in logic classes, we typically frame it as truth-preservation: a valid argument is one in which, if the premises are true, the conclusion must be true. And most of what is taught in such classes concerns truth-preservation, either directly or indirectly. However, as I've noted before, there is no fundamental distinction to be made between truth values and modal operators; the latter can be treated as a particular kind of truth value and the former as a particular kind of modal operator. Thus there is no good reason why logic can't concern itself with other kinds of preservation.
In a sense one can say that it has from the beginning. The centrality of demonstration in Aristotle's logical work can be said to make his primary concern not truth-preservation but necessity-preservation: he wants to know the preconditions for saying of an argument that if its premises are necessary the conclusion will also be necessary. Aristotle's famous definition of a syllogism ("A syllogism is a logos in which, certain things having been supposed, something different from these necessarily results because of their being so") has sometimes been taken to indicate that the premises and conclusion have to be relevant to each other, and that therefore Aristotle's logic is a kind of relevance logic; I am not sure that this is so, but it is clear enough that if you are interested in how arguments be necessity-preserving when the premises are different from the conclusion, you will be interested in something that can broadly be called 'relevance', because irrelevance is an impediment to necessity-preservation. One could also cash out the distinction between perfect and imperfect syllogisms in these terms: perfect syllogisms, those of the first figure, are those which make it obvious that if the premises are necessary the conclusion must be necessary; thus it makes sense to regard them as a sort of 'normal form' for demonstration. In any case, for purely structural purposes the distinction between truth-preservation and necessity-preservation is not one that makes a difference: every necessity-preserving inference, for any standard notion of necessity, will also be truth-preserving. But, since not every truth-preserving inference is necessity-preserving, if you are especially interested in getting necessary conclusions from necessary premises, it's a distinction that can support some logical weight.
If you can distinguish necessity-preservation from truth-preservation, you can also go in the opposite modal direction and distinguish truth-preservation from possibility-preservation. Indeed, the preservation of any sort of modality -- obligation, pastness, knownness -- can be investigated in its own right. Preservationist approaches to paraconsistency are essentially doing this: preservationism is the investigation of consistency-level-preservation in arguments, or, to put it in other terms, preservationism is the investigation of a particular sort of possibility-preservation. I'm not fully familiar with preservationism, but my guess is that the form of possibility it preserves is the capability for being true or, in other terms, degree of coherence.
In short, there are plenty of other things on the table besides truth-preservation: many validities beside the standard one.