I am trying to wrap my head around the following problem: I have three objects lined up horizontally, a $2$-sphere, a circle, and another $2$-sphere. It is the wedge sum $S^2 \vee S^1 \vee S^2$. I am trying to find the fundamental group of this space as well as the covering spaces. For the fundamental group, I believe that I can use van Kampen in the following manner:
$\begin{align*} \pi_1(S^2 \vee S^1 \vee S^2) &= \pi_1(S^2) * \pi_1(S^1 \vee S^2) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * \pi_1(S^1 \vee S^2) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * (\pi_1(S^1) * \pi_1(S^2)) \\ \pi_1(S^2 \vee S^1 \vee S^2) &= 0 * \mathbb Z * 0 \end{align*}$
Does this make sense?
I am still trying to work out how to find the covering spaces.