Where can I find a short and 'easy' peer reviewed paper on something from computability, decidability or complexity?
"By definition" recent research papers usually contain non-trivial results and they often (always?) require a deep knowledge of the argument in order to be well understood, so if you pick a paper at random you'll probably find a bunch of obscure signs in your hands :-)

So my suggestion is to look at "old" research papers (but not "too old"):


pick a particular argument/theorem that you think you have understood informally (or an argument that intrigues you);
read more about it on a theory of computation book or on Internet (for example start from Wikipedia and look at the references at the bottom, or search it on Google adding the keywords "introduction" or "lecture notes");
finally search the paper in which that argument/theorem was originally published and read it;


For example one of the fundamental basic results in computational complexity is the (deterministic) Time Hierarchy theorem. Very informally the theorem says that given more time, a Turing machine can solve more problems. You can start reading the Wikipedia article, then pick a lecture note with a more formal proof Googling around (for example pick this one by Arora). Finally when you think that you've understood the proof well download and read the original paper R. Stearns and J. "Hartmanis paper On the computational complexity of algorithms", 1965. You'll find that it contains even more details and is more formal.