Showing that $||f||_1 \leq 3^{-1/2}||f'||_p$ I am trying to show that $||f||_1 \leq 3^{-1/2} ||f'||_p$ for any $1 \leq p <\infty$ when $f$ is absolutely continuous.  Any help would be appreciated.
Also, $f:[0,1] \to \mathbb{R}$ with $f(0) = 0$.

This is false for $p=1$. Let $  f(x)=\min( x/\epsilon,1)$ for small $\epsilon>0$. Then $\|f'\|_1=1$ and $\|f\|_1>1-\epsilon$.  Counterexamples also exist for $1<p<2$.
However, the inequality is true $p\ge 2$. Its more precise  form  is 
$$\|f\|_1 \leq (q+1)^{-1/q} \|f'\|_p \tag1$$
where $q=p/(p-1)$ is the Hölder conjugate exponent to $p$. When $p\ge 2$, we have $(q+1)^{-1/q}\le 3^{-1/2}$. 
As you can imagine, the proof of (1) involves Hölder's inequality. I'll set it up for you: 
$$
\int_0^1 |f(x)|\,dx 
\le \int_0^1 \int_0^x |f'(t)|\,dx 
= \int_0^1 (1-t) |f'(t)|\,dx \le \dots
$$
