Given a manifold $M$, let $f:M\to M$ have the property that $f \circ f = id_M$, and that $f$ has no fixed points. Suppose there exists another manifold $N$ and a surjective local diffeomorphism $\pi:M\to N$ such that for all $y \in N$, there exists $x \in M$ such that $\pi^{-1}(y) = \{x,f(x)\}$.
I want to check that $\Omega^*(M) = \Omega^*_+(M) \oplus \Omega^*_-(M)$, where $\Omega^*_{\pm} = \{\omega \in \Omega^*(M) \lvert f^*\omega = \pm \omega\}$, but I am confused as to how to interpret $\Omega^*_+(M) \oplus \Omega^*_-(M)$, and how the information above even really helps in solving this problem.
My thinking is that for any $\omega \in \Omega^*(M)$, the above means you can find an $\omega_1 \in \Omega^*_+(M)$ and an $\omega_2 \in \Omega^*_-(M)$ such that $\omega = \omega_1 + \omega_2$, but I'm not sure. If anyone could help, that would be appreciated.
Your thinking is correct, but you also need to show that $\omega_1$ and $\omega_2$ are unique.
For uniqueness, suppose $\omega_1 + \omega_2 = \omega_1' + \omega_2'$ with $\omega_1, \omega_1' \in \Omega^*_+(M)$ and $\omega_2, \omega_2' \in \Omega^*_-(M)$. Note that $\Omega^*_+(M)$ and $\Omega^*_-(M)$ are closed under addition, so $\omega_1 - \omega_1' = \omega_2' - \omega_2 \in \Omega^*_+(M)\cap\Omega^*_-(M)$; this intersection is trivial because $f^*\omega$ cannot be equal to both $\omega$ and $-\omega$ unless $\omega = 0$. So $\omega_1 - \omega_1' = \omega_2' - \omega_2 = 0$ and hence we have $\omega_1 = \omega_1'$ and $\omega_2 = \omega_2'$.
So if such a decomposition exists, it is unique. How do we find a decomposition? Suppose $\omega = \omega_1 + \omega_2$ was a decomposition with $\omega_1 \in \Omega^*_+(M)$ and $\omega_2 \in \Omega^*_-(M)$. Then $f^*\omega = f^*\omega_1 + f^*\omega_2 = \omega_1 - \omega_2$, so
\begin{align*} \omega_1 &= \frac{1}{2}((\omega_1 + \omega_2)+(\omega_1 - \omega_2)) = \frac{1}{2}(\omega + f^*\omega) \in \Omega^*_+(M)\\ \omega_2 &= \frac{1}{2}((\omega_1 + \omega_2)-(\omega_1 - \omega_2)) = \frac{1}{2}(\omega - f^*\omega) \in \Omega^*_-(M). \end{align*}
So we see that any $\omega \in \Omega^*(M)$ has a decomposition $\omega = \frac{1}{2}(\omega + f^*\omega) + \frac{1}{2}(\omega - f^*\omega) \in \Omega^*_+(M)\oplus\Omega^*_-(M)$ and this decomposition is unique. That is, $\Omega^*(M) = \Omega^*_+(M)\oplus\Omega^*_-(M)$.