A toolbox for algebraic topology This question has a very general part and a rather concrete part.
General:
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).
Special:
The above is incredibly imprecise 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 Goerss/Jardine: simplicial homotopy theory.
 A: Re: your first question: As a beginning topologist, I've also been on the lookout for such a text. A book which has looked promising to me is A User's Guide to Algebraic Topology, which can also be found here on Google Books.
A: You might've (hopefully!) found the answer to your questions by now, but I just came across this post, and in case you're still interested, I find the book 'Algebraic Topology' by Allen Hatcher to be incredibly well-written and self-contained. It's given as a recommended textbook for a lot of elementary algebraic topology courses across many universities - you can find it online at http://www.math.cornell.edu/~hatcher/, together with some other books and links to lecture notes.
All the best,
A: In looking at the analogue of a fibration for squares of spaces  you should also look at the "fibrant" condition, which adds to all maps of the square being fibrations the condition that the map from the source of the square to the pullback of the two maps to the target is also a fibration. This condition extended to  the case of $n$-cubes is developed  in 
D. A. Edwards and  H. M. Hastings,   Čech and Steenrod homotopy theories with applications to geometric topology, Lecture Notes in Math., 542, Springer, Berlin, 1976
and also exploited in 
Steiner, Richard;  Resolutions of spaces by cubes of fibrations. 
J. London Math. Soc. (2) 34 (1986), no. 1, 169–176. 
A: The subject is really way too big (as are so many others of course).  I worry a lot about students not in Cambridge or Chicago or Stanford or other places where there are people with folklore at their fingertips.  For spectral sequences as a tool, there is a lot to be said for McCleary's guide.  Kate Ponto and I just published a book this year, More Concise Algebraic Topology, that may be usable for localizations and completions (just the old-fashioned localize or complete at a set of primes) and that also gives a reasonable start on model categories.  Even with that limited scope, the book is much longer than we would like: there were just too many basic details and tools not well enough documented in the literature. There are quite a few other books that go into one or another aspect of the subject (Goerss-Jardine, Neisendorfer, Strom, or, earlier, Whitehead), but it is not to be expected that a single source will cover the ground.
A: Re your second question, I don't have it in front of me but I believe you'll find this in Artin and Mazur's book "Etale Homotopy Theory", near the very beginning of the first section.
A: This is very late, but regarding your second question, in the case of fibrations, see Proposition 7.6.1 in Selick's "Introduction to Homotopy Theory". That section contains a useful discussion of exact sequences in homotopy theory. For the analogous statement for cofibrations to hold, additional assumptions such as simply connectedness are needed.
