Good source for a point set topological introduction to CW complexes?

Fritsch's and Piccinini's Cellular structures in topology (CUP); basically CW complexes are the subject matter of the book, which is exactly what you are looking for (it has exercises in it).

In my book Topology and Groupoids I took the view that you should first learn and get an intuition for the basic properties of identification spaces and adjunction spaces, and the properties of finite cell complexes, before going on to the CW-complex case, i.e. of cell complexes with an infinite number of cells. The book has lots of exercises on all these matters. In the infinite case, it is a question of arranging the topology so that as many as possible of the properties of the finite case go over to the infinite case. (The book is published outside an academic publisher to keep the price down, and as an experiment, so I have to advertise it myself.)

For an Introduction to the background of the use of groupoids in algebraic topology, see this paper.