With inner product $\langle f,g\rangle=\int_0^{2\pi}f(t)\overline{g(t)}dt$ on $L_2[0,2\pi]$ how do $||f||$, $||f||_2$ and $\langle f,f\rangle$ relate?
Firstly, $\|f\|_1$ is not the norm of $L_2[0,2\pi]$ by definition; $\|f\|_2$ is. Secondly, $\|f\|_1$ does not satisfy the Parallelogram law (see the section “Parallelogram law” and “Real and complex parts of inner products” of this wiki page), so it cannot define a inner product. Thus, for your question, the norm is definitely $\|f\|_2$.