Let me begin with something essentially obvious: students should learn to work with non-closed points. In practice, this means learning how to use them to simplify life.
Here are some suggestions as to how to do that:
(a) Explain that coherent sheaves are generically free, and use this to prove things like generic smoothness of varieties (by applying it to the tangent sheaf).
(b) Explain carefully the proof of Chevalley's theorem that the image of constructible is contstructible. (Note that this latter result has the advantage of being extremely useful, and also has likely not been covered in any form in a previous varieties course.)
Note also that one can dedcuededuce the Nullstellensatz from this result, which kills two birds with one stone. (See the discussion in this answer, and the notes of Mumford and Oda that are linked there.)
(c) one can beef up (a) by looking at say a fibration $X \to Y,$ and then looking at fibres over a generic point of $Y$, and then extending information to a n.h. of that point. Incidentally, it was the desirability of this kind of argument that first led Zariski to point out the importance of studying algebraic geometry over non-algebraically closed fields. For him, these non-algebraically closed fields were not $\mathbb Q$ or $\mathbb F_p$, but rather function fields of varieties (with the initial ground field being a good old fashioned algebraically closed field).
Examples like this last one can really help demystify not just the role of generic points, but also the role of non-algebraically closed fields. (In particular, they show that the latter are not just of interest in number theory. Zariski was certainly not a number theorist!)