We focus on the problems of sign changes and nonvanishing of Fourier coefficients of holomorphic cusp forms. Let ${\lambda }_f\left(n\right)$, ${\lambda }_g\left(n\right)$ and ${\lambda }_h\left(n\right)$ be the $n$-th Fourier coefficients of primitive holomorphic cusp forms $f$, $g$ and $h$ for $SL\left(2,\mathbb{Z}\right)$, respectively. Firstly, we study the behavior of the signs of the sequences $\left\{{\lambda }_f\left(n^j\right)\right\}$ with $j\ge 3$, $\left\{{\lambda }_f\left(n^i\right){\lambda }_g\left(n^j\right)\right\}$ with $i\ge 1,j\ge 2$, and $\left\{{\lambda }_f\left(n\right){\lambda }_g\left(n\right){\lambda }_h\left(n\right)\right\}$ in short intervals, and present some quantitative results for the number of sign changes for $n\le x$. Here we improve and generalize previous results. Moreover, we investigate simultaneous nonvanishing of $\left\{{\lambda }_f\left(n^j\right){\lambda }_g\left(n^j\right){\lambda }_h\left(n^j\right)\right\}$ with $j\ge 1$ in short intervals.