Since in 2007-2008 you evoked [ Class 24, ยง1.8, The problem with locally free sheaves] the equivalence between locally free sheaves and vector bundles on a scheme, the following point, potentially confusing for a beginner, could be mentioned.
A locally free sheaf $\mathcal E$ has a sheaf fibre $\mathcal E_x$ at $x$ but also a vector fibre $\mathcal E[x]=\mathcal E_x \otimes _ {\mathcal O_x} k(x)$. The fact that tensoring is not exact explains the paradox that a locally free subsheaf of a locally free sheaf does not yield a sub-vector bundle of a vector bundle in the above equivalence. The contrasting notation $\mathcal E[x]$ versus $\mathcal E_x$ (that I learned from German mathematicians) may help clarify this subtle point .
I am quite aware that there is nothing grandiose in this technical suggestion, but little points like those can be quite frustrating when learning a new subject
