The solid torus $X$ is the solid of revolution in $\mathbb R^3$ gotten by revolving the circle around the $z$-axis, in other words a donut. Its boundary $T$ is the torus.
Describe the homomorphism $i_* : \pi_1(T) \to \pi_1(X)$ between the fundamental groups induced by the inclusion map.
I really have no idea how to approach this problem, any help would be appreciated.
You certainly know there exists a homeomorphism $h: S^1 \times S^1 \to T$. See for example Finding the description of a homeomorphism from $S^1 \times S^1$ onto the surface of doughnut..
This homeomorphism extends to a homeomorphism $H: S^1 \times D^2 \to X$. Under this homeomorphism the inclusion $i : T \to X$ corresponds to the inclusion $J = id \times j : S^1 \times S^1 \to S^1 \times D^2$, where $j : S^1 \hookrightarrow D^2$.
For any two pointed spaces $(X_k,x_{0k})$ let $p_k : (X_1 \times X_2, (x_{01},x_{02})) \to(X_k,x_{0k})$ be the projection maps onto the $k$-th factor. It is well-known that the homomorphism $$P : \pi_1(X_1 \times X_2) \to \pi_1(X_1) \times \pi_1(X_2), P(c) = ((p_1)_*(c),(p_2)_*(c)$$ is an isomorphism of groups.
This yields the following commutative diagram: $\require{AMScd}$ \begin{CD} \pi_1(S^1 \times S^1) @>{P}>>\pi_1(S^1) \times \pi_1(S^1) @= \mathbb Z \times \mathbb Z @= \mathbb Z \times \mathbb Z \\ @V{J_*}VV @V{id _*\times j_*}VV @VV{id \times 0}V @VV{p}V\\ \pi_1(S^1 \times D^2) @>{P}>> \pi_1(S^1) \times \pi(D^2) @= \mathbb Z \times 0 @= \mathbb Z \end{CD} where $p(a,b) = a$. This should answer your question.