CQFT is very much still an open research subject. I don't think it is known what the best approach is. So all I can do is share my own opinion. (And a warning: I'm just an interested observer!)
First: Learn you some classical electrodynamics! Quantum field theory is after all a theory of fields, so one really should spend some time thinking about electrodynamics. It's the main example! Skipping it is a bit like skipping spheres to study homotopy categories.
Electrodynamics books tend to be a little cluttered, since they must also serve the needs of future engineers and experimentalists. You could skip around in Griffiths or Jackson, but I think the most efficient & logical approach to the core material of electrodynamics is Landau & Lifshiftz, Volume 2. Landau takes a straight route to Maxwell's equations, and then treats electro- and magnetostatics, EM waves, scattering and so forth as applications.
To your larger question: I think you should not attempt to read the CQFT literature without reading some of the physics literature as well. The CQFT literature doesn't exist in isolation, so studying it alone is at best only hearing part of a conversation. For another, most of the existing textbooks on CQFT are basically research monographs. You're expected to bring to them some intuition for the problem domain. (The exception to this is Dimock's Quantum Mechanics & Quantum Field Theory: A Mathematical Primer, which could only be better if it were longer.)
But the physics literature is very large! And written by people who either enjoy trolling mathematicians or are genuinely unable to distinguish between the $SO(3)$ and $\mathfrak{so}(3)$. So, yes, you have to be choosy. And how to be choosy is a matter of taste, so this is where I get opinionated.
A few suggestions, specifically about which physics books to read to gain intuition and context. (I'll let others try to make a map to topics of contemporary interest in CQFT.)
Avoid particle physics.
So many people have gone down this road, and it doesn't go anywhere! You just end up with mathematical translations of Peskin & Schroeder. Lots of calculational techniques, very few quantum fields. Frankly, I think this approach is basically doomed. Particle physics is:
too complicated: the details of the calculations obscures the basic structures.
not well understood: e.g., the infrared structure of even trivial theories like the quantum Maxwell theory is one of the major research themes of the past decade, never mind the problem of confinement.
probably broken at a foundational level: It's highly likely that neither quantum electrodynamics nor the Higgs field nor the Standard Model itself actually exist as continuum models.
So the easy examples don't exist, in the same sense that the translation invariant probability measure on the integers doesn't exist, and the harder examples are so hard you might win a Millenium prize. I'm not saying don't study particle physics, but don't make it your only focus.
Learn QFT via statistical physics
Take the other path: Learn QFT from the statistical physicists. It's easier! They have non-trivial solvable examples! In low dimensions! Also: the core organizing principles of QFT -- namely renormalization as a means of isolating the low-frequency behavior -- was discovered by people doing statistical physics (Kadanoff & Wilson), so you might as well follow along
Also, there's so much good mathematics here, from basic analysis and combinatorics in the classical theory of gases up to the recent work of Fields medalists like Hairer and Duminil-Copin.
If I were doing it over again, I'd probably read Goodstein's States of Matter together with Ruelle's more rigorous Statistical Mechanics: Rigorous Results. Then progress to Shankar's Quantum Field Theory and Condensed Matter: An Introduction. Also, Baxter's Exactly Solved Models in Statistical Mechanics.
Spend some time learning about QFT on curved spacetimes.
You learned all that differential geometry, so you might as well use it! And thinking about what QFT on curved spacetime should be is a good thing to do, since it forces you to focus on the fundamentals, since you can't just Fourier analyze everything in sight.
If you're a mathematician, the best thing to do here is to read Robert Wald's writings, starting with the little red book Quantum Field Theory in Curved Spacetime and Black Hole Thermodynamics. Wald is a very careful thinker and writer and he works with mathematics in very tasteful ways. (And I just learned, he's also written a grad level textbook on electromagnetism, which looks like a nice alternative to Landau.)
Even if you don't want to think about black holes and the like, you should make an effort to understand the Unruh Effect: Observers in different rest frames will agree on the value of a quantum field at a point, but they will not agree on how many particles are in a system.
Read lattice QFT papers
The physics literature on lattice QFT tends to have very clear definitions. They have to, because they're usually oriented towards numerical simulations, and you can't just hand-wave the details to a computer. It's also usually the starting point for rigorous CQFT constructions.
Start off with LePage's Lattice QCD for Novices. Then (if you still want particle physics), pick up Montvay & Munster's Quantum Fields on a Lattice.