I think the easiest method may be to use the expressions in terms of ladder operators (2.27) and (2.28) and the commutators in (2.32).
Alternatively, if you want to do the calculation in the position basis as Peskin & Schroeder do, you'll find the identity [A,BC]=[A,B]C+B[A,C] very useful. For example,
[\phi(\textbf{x},t),(\mathbf{\nabla}\phi(\textbf{x}',t))^2]= [\phi(\textbf{x},t),\mathbf{\nabla}\phi(\textbf{x}',t)]\cdot\mathbf{\nabla}\phi(\textbf{x}',t)+\mathbf{\nabla}\phi(\textbf{x}',t)\cdot[\phi(\textbf{x},t),\mathbf{\nabla}\phi(\textbf{x}',t)] \end{aligned}
Where [\phi(\textbf{x},t),\mathbf{\nabla}\phi(\textbf{x}',t)] can be easily calculated using (2.27).