I wish I could point to a single book that does what you want. The best I can offer is this suggestion: Read (or rather, skim) a few books on axiomatic field theory. Just get a sense for what the physics being described by the axioms is. Draw cartoons. Don't get too hung up on the topology.
I'd start with Haag's book, Local Quantum Physics. Learn about nets of algebras, which are a way of encoding the idea that physical observables actually are located somewhere in spacetime. Also does a nice job of explaining the idea that there's some sort of algebra of observables, and that states -- both pure states and density matrices -- are linear functionals on this algebra. Notice how crucial the fact that spacelike separated observables commute is to matching up the product of operators with the product of observables.
The thing really missing in Haag's book, if you're reading axiomatics as a way of organizing you're thinking about the material in some particle physics book, is a notion of field. The algebras in Haag's book really ought to be generated by 'local observables'. There should be operator-valued functions -- or rather, distributions -- on spacetime, and you should get the algebras in Haag's book by smearing these operators with test functions and then multiplying them. There is a language for this, which is nicely explained in Streater & Wightman's PCT, Spin, Statistics, and All That. This book also explains a critical reconstruction theorem: If you can write down all of the correlation functions of a QFT, then you can recover the Hilbert space & operators directly from these correlation functions.
Thinking of the basic local fields as generators of the algebra of observables is a good idea, but it's more complicated than you might think. The product of two fields at a point is in general singular. Equivalently, many of the correlation functions diverge as you approach the diagonals. You can define non-trivial products by subtracting away these divergences.
It's worth reading Hollands & Wald's paper Axiomatic Quantum Field Theory in Curved Spacetime, at about this point. They axiomatize QFT on globally hyperbolic spacetimes in terms of operator product expansions. Hollands and Wald's point of view is very close to the modern Wilsonian point of view on QFT, which says a) that a QFT is a deformation of a conformal field theory, and b) that a CFT is a collection of local observables obeying an OPE which satisfies the bootstrap condition. You could probably alter their diffeomorphism language to something involving conformal maps and get a definition.
It's also worth spending more time thinking about OPEs and CFTs. Spend some time with diFrancesco et al's Conformal Field Theory or maybe one of the math books on vertex algebras (Kac, Vertex Algebras for Beginners or Frenkel-Ben-Zvi Vertex Algebras & Algebraic Curves)?
Then at last, the path integral. It's not the most general definition of a QFT, but it's very useful. Basically, it gives you a way of writing down the correlation functions, and from the correlation functions, you can read off the OPEs, recover the Hilbert space, etc. The basic idea is best learned first in lattice QFT; I like Montvay & Munster's Quantum Fields on the Lattice. This is also a good time to learn a bit about gauge fields.
The biggest annoyance of the path integral approach is that the funcrtions on field space which are integrable with respect to the continuum measure -- the ones that represent observables in whatever vacuum you're playing in -- can be hard to express in terms of variables that people typically use to construct the measure. This is what renormalization is all about; it gives you a way of using stupid variables and then isolating the main contributions to the correlation functions you're after.
The standard mathematical reference on path integrals is Glimm & Jaffe, Quantum Physics: A Functional Integral Point of View. Most of what you'll want is in Chapter 3,6, or Appendix A. They spend most of the book grinding out a detailed construction of the correlation functions of 2d massive scalar field theory. Figuring out how their constructions are related to standard renormalization language is a worthwhile exercise.