I'm pretty much in your spot. I think part of the way there is learning to think with universal properties. I recently found a really good book (Algebra: Chapter 0, link below) on 'basic' algebra using category theory to unify things. All the basic stuff like products, disjoint union, surjections and injections are treated rigorously and in great generality through their universal properties. If you already know your group and set theory reading through the first few chapters can be done quickly, and should get you in the right mode of thought. I'm doing this myself right now, and so far I recommend you do the same.
EDIT: A nice application of the tensor product can be found in the first few pages of 'Differential Forms in Topology', that is, if $\Omega^*$ is the algebra generated by the formal symbols $dx_j,j=1,\dots,n$ under the relations $dx^2=0$ and $dx_idx_j=-dx_jdx_i$, then
$\Omega^(U)=C^\infty(U)\otimes\Omega^$ \Omega^*(U)=C^\infty(U)\otimes\Omega^*$ is the algebra of differential forms on the open set $U$ (under the wedge product). I'm not sure if that's how it's primarily used.(It seems $\LaTeX$ exploded on me.)