For an introduction to the basics of quantum field theory you could look into Introduction to Quantum Field Theory for Mathematicians. Lectures 13 and 18-22 introduce the $\phi^4$ model in 3+1 dimensions and the perturbative calculation of transition probabilities (from momenta $p_1,p_2$ to $p_3,p_4$). The first order term in the coupling constant is finite, but the second-order term diverges. Renormalization is then introduced to obtain a finite answer.
This perturbative approach to QFT is not mathematically rigorous, but you will obtain answers to some of the questions stated in the post (like "why add a $\phi^2$ term?" --- it gives the particles a mass).
For more rigour you then want to turn to the constructive approach to QFT. I understand your interest is in bosonic fields. For a broad overview you could take a look at A Perspective on Constructive Quantum Field Theory, to see that there exists a great variety of approaches in this category.
One rather recent development that I think requires the least amount of background is the Tree Quantum Field Theory of Gurau, Magnen, and Rivasseau. This a reformulation of the combinatorial core of constructive quantum field theory, which is neither based on functional integrals, nor on Feynman graphs, but on marked trees. For an application specifically to the $\phi^4$ model, see Constructive $\phi^4$ field theory without tears, by Magnen and Rivasseau.
One advantage of focusing on this modern approach is that it might be an entry point for original research (which may or may not be your objective).