Introductory algebra courses tend to systematically confuse products with coproducts, and more generally, confuse targets with domains. This systematically causes confusion in students (what is the difference between the two kinds of infinite product? and why are there two kinds? and how do I decide which to use when?).
Even if no category theory is going to be introduced, this terrible confusion should be eliminated.
In a related note, I regard it as an extremely important point, that should be celebrated, that for abelian groups, or vector spaces, etc., sums and products agree. In my experience, it is glossed over: "the sum and the product are the same, so don't worry about it; and I'll use the two notations interchangeably."
And another thing: the free product of two groups -- really?