A very brief review of an allergic response is in order so that the answer makes sense. Think of a linked chain of events here.
An "allergen" is something a person is allergic to, e.g., cat dander or pollen. When someone is exposed to an allergen, they either become allergic or not.
In a susceptible individual, something in the allergen (called an antigen, usually a protein of some kind) causes the person's immune system to produce an antibody (or Immunoglubin, in this case Immunoglobin E, or IgE) to the antigen, which then circulates throughout the entire system. The individual is now considered "sensitized". The IgE's bind to receptor sites on mast cell surfaces.
When a sensitized individual is re-exposed to the antigen, the antigen binds to the IgE's changing the cell surface, triggering a response.
Mast cells, found in connective tissues throughout the body - but especially in mucus membranes and skin - make and store a molecule called histamine. When the antigen binds to the IgE's on the mast cell, the cell is "signalled" to release (among other things) histamine, a molecule which causes local capillaries to swell and leak fluid: the typical runny/stuffy nose, red/itchy/watering eyes, etc.
So that's a basic allergic response involving histamine.
Is the antihistamine doing something regarding his immune's system reaction or is it doing something about the symptoms? How does it work?
The antihistamine essentially blocks the effect of histamine on the capillaries. It doesn't decrease his immunity to the antigen (it doesn't stop IgE from being produced); it just treats the symptoms (stuffy nose, watery eyes, etc.)
Will his immune system learn how to behave whilst taking antihistamines? Would the process of learning be better or worse for taking them?
The antihistamine will have [little or]* no effect - either positive or negative - on his immune system. It is used only to make the allergic symptoms more tolerable.

*The immune system is so incredibly complex, many things are still being worked out. I would not categorically rule out some kind of modulation, but for all intents and purposes, this is not what antihistamines are used for.
Image from The Asthma Center
Histamine and H1-antihistamines: Celebrating a century of progress
IgE and mast cells in allergic disease
Janeway, Immunobiology: The Immune System in Health and Disease. 5th edition
Histamine and H1-antihistamines: Celebrating a century of progress