You're talking about a pretty massive field. That being said The Nature of Mathematical Modeling by Neil Gershenfeld might be what you're looking for. Each chapter is a short (15-25 page) introduction to one common tool in applied mathematics. In that space you'll get an overview of when you might want to use that technique, a brief description of the basic mathematical theory underlying the method and a few examples of how it can be applied.

Of course you won't learn enough about any topic to be able to actually tackle any but the most trivial of problems, but you will come away with a good understanding of which technique you should use for which kind of problem and have a good enough of a basis to be able to understand the more advanced books on the given topics. Each chapter also ends with a recommendation of 3-4 books which you can check out if you want to dig deeper into that particular topic.