I'm not an expert in this topic too, but I'm trying wrap my head around it.
Right now I'm trying to make an adequate hierarchy of concepts related to renormalization. Let me list them and tell how they are related:
- Fields, Lagrangian (Hamiltonian) and coupling constants.
- Perturbative calculations.
- Different scales.
- Quantum fields.
- Ultraviolet divergences.
- Renormalization group and running couplings.
(Let me stress that the "renormalization" and "renormalization group" are different concepts.)
Of course the concept of a field and the way to describe (1) it is a starting point.
Now, It seems to me (while I can be wrong) that every time we talk about renormalization we always deal with some perturbative approach (2). There is always something that we want to neglect. And if there is a way to make calculations without any approximations then one needn't to use techniques related to renormalization.
One of simplest examples is a hydrodynamics -- you don't want to "get down" to the level of molecules to describe a stream of water. You would like to work with some "integral" quantities, like viscosity. And the viscosity can be used to describe processes at many different scales (3): bloodstream, butterfly, submarine, internals of the star, e.t.c.
The hydrodynamics works at different scales because of the self-similarity (4): by going several orders of magnitude larger you are still able to describe your system with the same Lagrangian, but, maybe, with some parameters changed. When one does the transition from one scale to another one always neglects some peculiarities (2), that occur at smaller scale.
This is the essence of renormalization group(8) techniques. The changing parameters are also called the running couplings. I recommend you to read about Kadanoff transformation, to get more insight about it.
Note that I never mentioned divergences by far. Because this is a slightly different topic. And one can use renormalization group even if there is no infinities.
UV divergences appear due to our ignorance about smaller scales. When we talk about hydrodynamics we know that there is a "fundamental scale" -- the aforementioned molecules. But when we talk about quantum fields (6) (like electromagnetic field or some fermion field) we don't know what is the "fundamental" scale for it. We don't even know if it exists at all.
Different methods of dealing with the divergences are called the renormalization (7) methods. They are based on changes of the parameters of Larangian too, but now these changes are "infinite" because one have to "compensate the infinities" appearing from small scales. After cancelling the infinities this way one is still left with arbitrariness of choosing a finite values of the parameters. You can fix the parameters by getting them from experiment at certain scale(3) and use renormalization group (8) to go from one scale to another.