We consider a two parameter family of generalized Ces\'aro operators $P^{b,c}$, $Re(b + 1)>$Re$c>0$, on classical spaces of analytic functions such as Hardy ($H^p$), $BMOA$ and $a$-Bloch space ($B^a$). We Prove that $P^{b,c}$, $Re(b + 1)>$Re$c>0$ is bounded on $H^p$ if and only if $p\in(0,\infty)$ and on $B^a$ if and only if $a\in(1,\infty)$ and unbounded on $H^{\infty}$, $BMOA$ and $B^a$, $a\in (0,1]$. Also we prove that $\alpha$-Ces\'aro operators $C^{\alpha}$ is a bounded operator from the Hardy space $H^p$ to the Bergmann space $A^p$ for $p\in (0,1)$. Thus, we improve some well known results of the literature.