"I want to step-up my printing: should I start with fiber paper?"
In a word, no. You should be able to make really good prints with RC paper. At a company I worked for many years ago we would receive product prints on B&W RC paper from the advertising agency we used. They never failed to impress me. Of course there is no reason not to try fiber based paper. I'm kind of surprised that you haven't yet.
There is a thread somewhere about recommended printing books. There should be lots available cheaply on the used market. If you are anywhere near the Chicago area, there is a very good amateur club that has open printing sessions with the members.