Falling into the Schwarzschild black hole. Important details

The Schwarzschild space is one of the best studied spacetimes and its exhaustive considerations are easily accessible. Nevertheless, by some reasons it is still surrounded by a lot of misconceptions, myths, and "paradoxes". In this pedagogical paper an attempt is made to give a simple (i. e., without cumbersome calculations), but rigorous consideration to the relevant questions. I argue that 1) an observer falling into a Schwarzschild black hole will \emph{not} see "the entire history of the universe" 2) he will \emph{not} cross the horizon at the speed of light 3) when inside the hole, he will \emph{not} see the (future) singularity and 4) the latter is \emph{not} "central".