Since the function $f(x) = \begin{cases} 2^{x}, & x < 1 \\ -x+3, & x\geqslant 1 \end{cases}$,

First, we find $f(0)$. According to the function definition, when $x<1$, $f(x) = 2^{x}$. So, $f(0) = 2^{0} = 1$.

Next, we find $f[f(0)]$, which is equivalent to finding $f(1)$. According to the function definition, when $x\geqslant 1$, $f(x) = -x+3$. So, $f(1) = -1+3 = 2$.

Therefore, the value of $f[f(0)]$ is $\boxed{2}$.

This problem tests our understanding of how to find the value of a function at a given point. It's important to carefully read the problem and make reasonable use of the properties of the function.