I think the best way to appreciate abstraction is to actually see examples of vector spaces which are not $\mathbb{R}^n$ in any obvious way. For instance all polynomials of degree $k$ such that $p(0) = 0$ or all symmetric $3 \times 3$ matrices. For a more subtle example, subset of a finite set $X$ are a vector space over $\mathbb{Z}/(2)$ taking as $+$ the symmetric difference (until one realizes that this is just $(\mathbb{Z}/(2))^n$ in disguise). Students should be offered many exercises with these vector spaces, so that they become familiar.
When finite dimensionality is not necessary, one can make even better. For instance it is very nice to see the derivative as an example of a linear operator, and if one wants to have a finite dimensional example one can take the vector space of all solutions to some constant coefficients linear differential equation. Even a particular one, like $4f''' + 2f'' -f' -2 = 0$ will do. The derivative of a solution is a solution, hence we have a very natural linear endomorphism of this space. And by the way some linear algebra (for instance Jordan decomposition or even less) can be used to actually solve the equation.
Moreover I think that the fact that all vector spaces are isomorphic to $\mathbb{R}^n$ shows the power of abstraction at its best. Geometrically we only need to have the intuition for one very simple case; but then the proofs we give will apply to a plethora of other unexpected cases.