Consider the $2$-torus $T^2 = S^1 \times S^1$ and consider the space $X = T^2/(S^1 \times \{1\})$, $T^2$ with $S^1 \times \{1\}$ collapsed to a point. What CW-complex structure does $X$ have and how would we compute its fundamental group?
I am not sure how to think about this. It seems to me that $X$ would just be homeomorphic to a circle, but I am not sure.
This is almost a comment, but one cannot post images as comments.
Here is an image of what the space looks like. This should help you in finding a suitable CW-structure. Also, this topological space is a called a pinched torus.
Also, the space is homotopy equivalent to the wedge sum of $\mathbb{S}^2$ and $\mathbb{S}^1$ as illustrated in Hatcher Example 0.8. WIth this one can compute the fundamental group of the space to be $\mathbb{Z} (= \{0\} \ast \mathbb{Z})$.