As David and Anweshi told before, think it could be very interesting to deal with functor of points, with main example being subfunctors of Grassmannians. I would make some general statements on functor of points (Yoneda lemma, definition of functor of points, vector bundles) and then begin to study as soon as possible classical examples, such as Grassmanians, Severi-Brauer varieties and their tautological vector bundle, varieties of flag of subspaces...
Finally it would lead to a glimpse on group schemes and algebraic groups.
