I would suggest the approach Tom Apostol takes in his linear algebra book. In chapter 1, after introducing abstract vector spaces, he goes on to Gram-Schmidt, and then immediately to best approximations. At the end of the first chapter, he solves questions like: "find the polynomial of degree three $p(x)$ which approximates $\sin(x)$ best over $[2,3]$ in the sense of minimizing the error $\int_2^3 (sin(x)-p(x))^2 dx$.
When I first read this, I was amazed. Prior to this, I only knew high school mathematics plus basic calculus - no abstract math at all. The problem of approximating one function by another seemed completely unsolvable given the mathematics I knew at the time. And yet here it had a simple solution.
Even more amazingly, the solution was right in front of me all along. If you had asked me how to approximate the vector $(1,2,3)$ by a vector whose last coordinate was $0$ - I would have immediately said $(1,2,0)$. I knew a little bit about geometry problems, and the problem of finding the closest point in a plane seemed "easy" and "natural" to me. And yet this this is all the solution of this problem required - all I needed was just to think about "vectors" or "points" a little more abstractly. I was completely sold on the benefits of the abstract approach.