Consider the space $$Y=\{(x_1,x_2,x_3):x_1^2+x_2 ^2+x_3 ^2=1\}\cup \{(x_1,0,0):-1\leq x_1\leq 1\}\cup\{(0,x_2,0): -1\leq x_2 \leq 1\}.$$ So $Y$ is essentially the unit sphere together with the segments of the $x$ and $y$-axes inside the sphere.
I am trying to calculate $\pi_1(Y)$ of this space; I am aware that I may have to use van Kampen's theorem along the way, but I am currently stuck on how to proceed.
Any help will be very useful. Thanks in advance.
Take a look at this answer to convince yourself that your space $Y$ is homotopy equivalent to the space $S^2 \vee S^1 \vee S^1 \vee S^1$. Then you can use Seifert-van Kampen to show that $\pi_1(Y) \cong F_3$, i.e. the free group on three generators. I hope this helps!