Vivid examples of vector spaces?
The real vector space of all fibonacci sequences (the first two values are arbitrary) is quite instructive. Or the subspace of all smooth functions satisfying the differential equation $f'=f$. It is quite illuminating that elementary linear algebra has a interplay with analysis.
The positive real numbers, where 1 is the "zero vector," "scalar multiplication" is really numerical exponentiation, and "addition" is really numerical multiplication.
Simplicial complexes, as in algebraic topology, are another good example, but perhaps this is even too weird. Still, it might be fun to throw out the idea that mathematicians like to add triangles to each other to get quadrilaterals and negate them to reverse orientation, but they'll probably have to take it on faith that this is actually useful.
I like the example $C([0,1])$ of continuous functions on the interval (or something similar). It is familiar-looking but shows that there is not always a natural choice of basis.
I like the color example. It shows how the idea of a basis is useful, even though it's not a vector space.
Barycentric coordinates are another example of something like a vector space but not a vector space.