I find that most books on algebraic topology make things more difficult for students than seems necessary by not using paths of "arbitrary length" so that the paths under composition form a category, that is composition is associative and each path has a left identity and a right identity. That is one can define a path (of length $r$) for some $r \geqslant 0$ in $X$ to be a map $f: [0,r] \to X$; or to be a pair $(f,r)$ where $r \geqslant 0$ and $f: [0, \infty) \to X$ is constant on $[r, \infty)$.
Second the notion of the fundamental groupoid $\pi_1(X,A)$ of $X$ on a set $A$ of base points was introduced by me in 1967, and has many advantages over the usual fundamental group. See my answer to this question: http://mathoverflow.net/questions/40945/compelling-evidence-that-two-basepoints-are-better-than-one
This tool allows more powerful theorems with in many cases simpler or clearer proofs, and is developed and applied in my book Topology and Groupoids, the 2006 edition of a book published in 1968; this groupoid $\pi_1(X,A)$ is used in no other topology text in English, to my knowledge. See also this downloadable book Categories and Groupoids.