In quantum field theory we have the concepts of regularization and renormalization. I'm a little confused about these two. In my understanding regularization is a way to make divergent integrals convergent and in renormalization you add terms to the Lagrangian which in turn cancel the divergent integrals.
Is there a connection between the two of these, or are these two separate ways of dealing with the infinities, i.e. do you use them both together to make the divergences disappear?
Regularization and renormalization are conceptually distinct.
As you essentially indicate, regularization is the process by which one renders divergent quantities finite by introducing a parameter $\Lambda$ such that the "original divergent theory" corresponds to a certain value of that parameter. I put "original divergent theory" in quotations because strictly speaking, the theory is ill-defined before regularization.
Once you regularize your theory, you can calculate any quantity you want in terms of the "bare" quantities appearing in the original lagrangian (such as masses $m$, couplings $\lambda$, etc.) along with the newly introduced regularization parameter $\Lambda$. The bare quantities are not what is measured in experiments. What is measured in experiments are corresponding physical quantities (the physical masses $m_P$, couplings $\lambda_P$, etc.).
Renormalization is the process by which you take the regularized theory, a theory written in terms of bare quantities and the regularization parameter $(\Lambda, m, \lambda, \dots)$, and you apply certain conditions (renormalization conditions) which cause physical quantities you want to compute, such as scattering amplitudes, to depend only on physical quantities $(m_P, \lambda_P, \dots)$, and in performing this procedure on a renormalizable quantum field theory, the dependence on the cutoff disappears. So, in a sense, renormalization can be thought of as more of a procedure for writing your theory in terms of physical quantities than as a procedure for "removing infinities." The removing infinities part is already accomplished through regularization.
Beware that what I have described here is not the whole conceptual story of regularization and renormalization. I'd highly recommend that you try to read about the following topics which give a more complete picture of how renormalization is thought about nowadays:
You may also find the following physics.SE posts interesting/illuminating:
What exactly is regularization in QFT?
Regulator-scheme-independence in QFT
It's exactly like you said. Regularization is a mathematical procedure, with it, you can separate and isolate the divergence of the integrals. Regularization techniques include: dimensional regularization, dimensional reduction, implicit regularization, and others. You can read more about them in here. On the other hand, renormalization is how we absorb the divergence in the parameters of the theory. So, renormalization is related to the physics and how to understand the nature of the divergence.
