As Igor said, it's a bit late for that. If you just finished undergrad and you discovered a passion for QFT from a rigorous mathematical standpoint, what you should do, for example, is apply for the math PhD program at UVa and then do a thesis with me ;) Without expert help, trying to get into the subject by reading material in topology, differential geometry or algebraic geometry, as preparation for introductory courses that should hopefully equip you with the necessary mathematical background to finally be ready to start thinking about a research problem related to rigorous QFT, sounds like a recipe for not getting anywhere.
That being said, if you are serious about your goal, here is what you can do. Pick one of the QFT models that have been constructed rigorously and study that proof of existence until you understand it completely. I would recommend a result where the method is sufficiently general so by learning an example you actually get a feel for the general situation. This is in line with Hilbert's quote about the example that contains the germ of generality. In the present situation, this narrows the pick to a proof of construction of a QFT model using renormalization group methods (If you wonder why, see edit below).
As a rule, Fermionic models are considerably easier that Bosonic models, when it comes to rigorous nonperturbative constructions. I therefore think the best pick for you would be the article "Gentle introduction to rigorous Renormalization Group: a worked fermionic example" by Giuliani, Mastropietro and Rychkov. It is pretty much self-contained. If you know the Banach fixed point theorem, you're in business. In the article, they construct an RG fixed point, which in principle corresponds to a QFT in 3d which conjecturally is a conformal field theory. What they do not do is construct the correlations from the knowledge of that fixed point. As a consequence, they also do not prove conformal invariance of correlations. So here are two contemporary research problems for someone who did the "homework assignment" I just mentioned and would like to go further and prove something worthwhile.
If you prefer Bosonic models, then the other pick I would recommend is the article "Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions" by Chandra, Guadagni and myself. It is a toy model for the Bosonic analogue of the example considered by Giuliani, Mastropietro and Rychkov. The spacetime on which the fields are defined has a hierarchical structure which facilitates the multiscale analysis. There we rigorously constructed the RG fixed point and also the correlations for two primary fields, the elementary scalar field and its square. What we did not prove is conformal invariance, which is also conjectured to hold. The definition of conformal invariance in this setting is the same as in my previous answer:
What is a simplified intuitive explanation of conformal invariance?
Namely, just change Euclidean distance to the maximum of the $p$-adic absolute values of the components. Our article is also self-contained and also needs the Banach fixed point theorem together with some complex analysis and very minimal knowledge of $p$-adic analysis. The basics of $p$-adic analysis needed take a weekend to learn.
Edit addressing the OP's three new questions in the comments:
Why didn't I mention say TQFT or other approaches? Topological QFTs (the stress tensor vanishes completely) is a small subset of Conformal QFTs (the trace of the stress tensor vanishes) which themselves form a tiny subset of general QFTs. The study of these particular cases is certainly interesting but this relies on different tools that are specific to these particular cases and once you invest in learning these tools you will most likely be stuck with these particular cases for the rest of your research career. I proposed RG methods because I believe they cast a wider net and also should broaden your understanding of the subject. I think it would be easier to later specialize in say TQFT if that is where your taste leads you, rather than go the other way around: first develop expertise in say TQFT, and then learn some other method like the RG in order to escape from the narrow realm of TQFTs and study QFTs which are not topological. Next, a comment on "does this mean that all other method than rigorous renormalization group method have failed currently to construct a sensible QFT satisfying Wightman’s axioms (or its equivalent)?". The quick answer is no, this is not why I said you should choose an article which uses RG methods. I wrote my answer not just for you but also for other young people interested in rigorous QFT, from the hypothetical perspective of a PhD advisor talking to a beginning PhD student, starting the PhD thesis work now in 2021. As far as what method has been successful in proving the Wightman axioms for specific models, there are several. RG is one, as in the work of Glimm, Jaffe, Feldman, Osterwalder, Magnen and Sénéor on $\phi^4$ in 3d, and the work of Feldman, Magnen, Rivasseau and Sénéor for massive Gross-Neveu in 2d. The earlier work of Glimm, Jaffe, Spencer on $\phi^4$ in 2d used a single scale cluster expansion. Methods based on correlation inequalities and the more recent ones based on stochastic quantization typically allow you to prove most of the Osterwalder-Schrader axioms but not all, e.g., not Euclidean invariance.
Suppose we were having this discussion about geometric invariant theory instead of QFT, would you be asking me: I understand that Hilbert is the pioneering figure in the field of GIT, shouldn’t I start with some classical work at first? Sure, you could go read his 1893 Ueber die vollen Invariantensysteme but this might be more appropriate for a PhD in the history of math, rather than for doing research in this mathematical area now in 2021.
More or less yes, although I do not like your choice of words when you said that Axiomatic QFT "put QFT on the rigorous mathematical ground" in contrast to Constructive QFT which merely tries to "propose actual QFT models which satisfy those axioms". The ones who propose models are theoretical physicists. They come to you and say: here look at this Lagrangian it describes a model which is important for physics. Then you, say the constructive QFT mathematician, your job is 1) to prove that this model makes sense rigorously, by controlling the limit of removing cutoffs with epsilons and deltas but certainly no handwaving, and 2) to prove the limiting objects satisfy a number of properties like the Wightman axioms. Then the axiomatic QFT person can come and say: as a consequence of satisfying the axioms, here are these other wonderful properties that your model also satisfies, by virtue of this general theorem I proved the other day. Hope this clarifies the logical articulation of these different subareas of rigorous QFT.