This question has a very general part and a rather concrete part.
When one wants to prove something in algebraic topology (actually in all parts of mathematics) one obviously needs some good ideas, but first one has to have a good set of tools at hand. Introductory books in algebraic topology provide a number of such tools like long exact sequences to name just one. If one proceeds working in that field and reaches research level more and more tools are just treated as "common knowledge". They are used in papers according to the current situation and often left without quotation.
Over the time one gathers plenty of those tools, but I for my part still take many of them as black boxes. When I use them I always have the feeling of walking on very thin ice. Most advanced books have some of those tools scattered in their body and finding a particular one is often harder than it should be. There they are used to build up a certain theory and often don't reveal themselves as useful tools with applications beyond the topic of the respective book. Moreover it is one thing to find the reference for a statement one knows to be more or less true, but realising which tool one has to use when one isn't even aware of the precise statement is a different story.
So the first question:
Are there any good books which provide a box of tools used in modern algebraic topology? Maybe something like "AT for the working mathematician".
They should come with a proof but not necessarily with applications (for the above reason).
The above is incredibly unprecise and there are so many ways to interpret the question. Hence one example of a statement I actually want to know about, which might also give a hint at what I am looking for.
Second question: What is the precise statement/where can I find a proof
Given a commutative square of fibrations (cofibrations). Then the fibers (cofibers) in the horizontal direction are homotopy equivalent if and only if the fibers (cofibers) in the vertical direction are homotopy equivalent.
The square is then cartesian, cocartesian, bicartesian?
Edit 1: Now that I think about it it looks like the second question is just an application of the snake lemma. I have to work out the details. Still this statement may stand as an example of what I am looking for.
Edit 2: A book which seems to go in the direction of what I describe might be Goers/Jardine: simplicial homotopy theory.