From Husemoller, page 8 (rephrased):
Theorem: Let $E, B \in \mathbf{Top}_*$, and $F \to E \to B$ be a Serre fibration. Then there is a natural group homomorphism $\partial: \pi_n(B) \to \pi_{n-1}(F)$ such that the sequence $\ldots \to \pi_n(E) \to \pi_n(B) \ \xrightarrow{\partial}\ \pi_{n-1}(F) \to \pi_{n-1}(E) \to \ldots$ is exact.
Exercise: apply this theorem to:
1. $\mathbb{Z} \to \mathbb{R} \to S^1$,
2. $\mathbb{Z_2} \to S^n \to \mathbb{R}P^n$,
3. $S^1 \to S^{2n+1} \to \mathbb{C}P^n$.
How to approach this? The definition via homotopy lifting is not very helpful. Husemoller mentions that it is sufficient to check each individual cell in a CW complex, but does it mean that I only have to check for $I^k,\ k = 1,\ldots,n$ [fixed bad typo]? Some hint would be nice, I'm not very familiar with CWs besides what Husemoller himself mentioned earlier (although I did make sure I understood all the proofs he gave).
PS: I asked for a lot of hints recently, it worries me. Were these questions hint-worthy?