After days of pondering about this type of problem, I can't came up with nothing.
Let $A=\{(0,1)\}\times S^2$ and $B=S^1\times\{(0,0,1)\}$. Define $X:=(S^1\times S^2)\setminus(A \cup B)$ and find $\pi_1(X)$.
In general, when dealing with this type of problems, I look for algebraic tricks. So, knowing that $$Y\setminus(Z\cup W)=(Y\setminus Z)\setminus W$$ I can observe that
$1)\,\, (S^1\times S^2)\setminus(A \cup B)=[(S^1\times S^2)\setminus A]\setminus B=[(S^1\setminus \{(0,1)\}) \times S^2]\setminus (S^1\times\{(0,0,1)\}),$
or alternatively,
$2)\,\,(S^1\times S^2)\setminus(A \cup B)=[(S^1\times S^2)\setminus B]\setminus A=[S^1 \times (S^2\setminus \{(0,0,1)\})]\setminus (\{(0,1)\}\times S^2).$
The idea I am following is to write the space as a product $U\times V$, so that knowing the fundamental groups of $U$ and $V$ I can compute $\pi_1(X)$. In this case, even knowing that $S^n\setminus\{N\}\approx \mathbb{R}^n$, I can't conclude.
Please note that I am searching for a general strategy to visualize topological spaces in higher dimensions.
Looking for algebraic tricks without the visualization is unlikely to succeed. In particular, attempting to impose unmotivated algebraic structure on your space such as a product $U \times V$ is more than likely going to send you down a long, pointless wandering path, given that product structures are so special.
Instead, let's look at these two sets, and start simple: $$A = \{(0,1)\} \times S^2 \qquad B = S^1 \times \{(0,0,1)\} $$ Each of these is a subspace of $\mathbb R^2 \times \mathbb R^3$.
Let's introduce some coordinates $((x,y),(r,s,t))$ for a point in $\mathbb R^2 \times \mathbb R^3$ (and let's also resist rewriting this as $\mathbb R^5$, for reasons that will be clear).
The first thing I notice is that the set $A$ is a subset of $\{(0,1)\} \times \mathbb R^3$, i.e. it is parallel to $r,s,t$-coordinate 3-space. Also set $B$ is a subset of $\mathbb R^2 \times \{(0,0,1)\}$, i.e. it is parallel to the $x,y$-coordinate 2-space.
What is the intersection of these two coordinate subspaces?
Now I use some high dimensional visualization. In 2-dimensions, any line parallel to the $x$-axis, intersected with any line parallel to the $y$-axis, is a single point. This generalizes very easily to $\mathbb R^m \times \mathbb R^n$: for any $X \in \mathbb R^m$ and any $Y \in \mathbb R^n$ we have $$(\{X\} \times \mathbb R^n) \cap (\mathbb R^m \times \{Y\}) = \{(X,Y)\} \subset \mathbb R^m \times \mathbb R^n $$ In particular, in $\mathbb R^2 \times \mathbb R^3$, the subspace $\{(0,1)\} \times \mathbb R^3$ intersected with $\mathbb R^2 \times \{(0,0,1)\}$ is a single point, namely the point $$P = ((0,1),(0,0,1)) $$ So, we've reached an interesting conclusion: the intersection $A \cap B$ is either the single point subset $\{P\}$ or the emptyset. But if we observe that $(0,1) \in S^1$ and that $(0,0,1) \in S^2$ then we reach the conclusion that $$A \cap B = \{P\} $$ From this, we can reach one more conclusion: $A \cup B$ is the wedge sum of $A$ and $B$.
And now we can apply a useful tool: by application of Van Kampen's Theorem, $$\pi_1(A \cup B,P) = \pi_1(A,P) * \pi_1(B,P) $$ In words, our group $\pi_1(A \cup B,P)$ is a free product of the two groups $\pi_1(A,P)$ and $\pi_1(B,P)$.
I presume you have the tools to finish this problem off, so I'll leave the pleasure of working out the final answer to you.