The formal definition states that a point $x$ is a limit point of $S$ unlesss there exists a neighborhood of $x$ that doesn't contain any points of $S$ (except, possibly, $x$ itself). Intuitively, if there are elements of $S$ that are really close to $x$, then $x$ is a limit point.
Let's handle your examples one at a time, starting with #3. The point $0$ is a limit point of $S$, because there are points in the open interval $(0,1)$ that are extremely close to $0$. Similarly, $1$ is a limit point, along with everything in $S$ itself. However, $17$ for example is not a limit point, because the neighborhood $(16, 18)$ contains nothing from $S$. This should be intuitively clear.
#2 is more difficult (for some reason, your textbook seems to put the harder problems first!), but you should be able to handle it if you consider the cases where $n$ is even and odd separately. Graph the points if you aren't sure.
#1 is less clear still. Hint: you should be able to show that $S$ is dense in $R$. You might want to use the fact that there are infinitely many primes.
Comment if you want any more help!