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 mathemetical background to finally be ready to start thinking about a research problem related to rigorous QFT, sounds like a recipe for not getting anywhere.

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.

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 <a href="https://link.springer.com/article/10.1007/JHEP01(2021)026">"Gentle introduction to rigorous Renormalization Group: a worked fermionic example"</a> 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 <a href="https://arxiv.org/abs/1302.5971">"Rigorous quantum field theory functional integrals over the p-adics I: anomalous dimensions"</a>
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:

https://mathoverflow.net/questions/394335/what-is-a-simplified-intuitive-explanation-of-conformal-invariance/394363#394363

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 and some complex analysis and very minimal knowledge of $p$-adic analysis.
The basics of $p$-adic analysis needed take a weekend to learn.