Let $p : E \to B$ be a Serre fibration with $b_0 \in B$ and $F=p^{-1}(b_0)$ and $e_0 \in F$. I have been trying to understand how this induces a long exact sequence $$\dots\xrightarrow{\quad \partial\quad}\pi_n(F,e_0) \xrightarrow{\quad i_*\quad}\pi_n(E,e_0) \xrightarrow{\quad p_*\quad}\pi_{n}(B,b_0)\xrightarrow{\quad \partial\quad}\pi_{n-1}(F,e_0)\xrightarrow{\quad i_*\quad}\dots$$
but I don't understand the maps in question. I'm also aware that this is covered in many books on algebraic topology which I've looked at, but it seems that many authors have drastically different ways to define these and I'm quite confused.
If I understood correctly $i_*:\pi_n(F,e_0) \to \pi_n(E,e_0)$ is induced from the inclusion $i:F \to E$ so that $i_*([\alpha])=[i\circ \alpha]$? Here $\alpha : (I^n,\partial I^n) \to (F,e_0)$ and I guess $i\circ \alpha$ makes sense?
The map $p_*$ is probably defined also like this. The main difficulty is the boundary map. I have no idea how it is defined in this case. I know how its defined in singular homology, but this isn't similar to that.