I am looking for a general reference text on model categories, that contains all the basic results and definitions. I'm perfectly happy to be pointed towards a textbook, and I'm not looking for something with modern fancy theorems, just standard results. I know the definition of "model category", but not much else. I have some experience with one or two particular model categories, and I can prove any result I really need "by hand" for my particular examples, but much better would be to cite standard facts than to reprove special cases of them.
Here's the type of fact that I'd like to find in such a reference (if it is in fact true). By this I mean also that the reference should include all necessary definitions, since I'm not even sure how to make precise the following claim:
Suppose that $A$ is a cofibrant object, and $\hat B \to B$ is an acyclic fibration. For each $f: A \to B$, the space of lifts $\hat f : A \to \hat B$ covering $f$ is contractible.
The definition I know just guarantees that it is non-empty, but surely contractibility (if correctly defined) follows. Anyway, where can I find this and similar results? Something like "Model Categories for the working mathematician"?