Although I know very little category theory, I really do find it a pretty branch of mathematics and consider it quite useful, especially when it comes to laying down definitions and unifying diverse concepts. Many of the tools of category theory seem quite useful to me, such as Mitchell's embedding theorem, which allows one to prove theorems in any abelian category using diagram chasing. It lets me the ability to treat lots of objects I would not otherwise feel comfortable with as if they were modules over some ring; in essence, I feel like I've gained some tools from it.
However, I simply cannot see where to apply the Yoneda lemma to some useful end. This is not to say that I don't think it is a very pretty lemma, which I do, or that I do not appreciate the aesthetic of being able to study an object in a category by looking at the morphisms from that object, which I also do. And I do find it useful to consider the modules over a ring rather than the ring itself when studying that ring, or to treat groups as subgroups of permutation groups, which are the two applications I've heard of the Yoneda lemma. The problem is that I already knew these things could be done. Essentially, I don't feel like I've gained any tools from the Yoneda lemma.
My question is this: how can the Yoneda lemma be applied to make problems more approachable, other than in cases like those I have listed above which can easily be treated without a general result like the Yoneda lemma? Basically, what new tools does it give us?