You don't get more algebraic geometry than understanding the equations defining a (affine or projective) variety, and there are still tons of questions being actively studied. For example:
Google "set-theoretic complete intersections". While you are at it, also search for "monomial curves".
"Secant varieties". These even have connections to statistic, etc, search for "tensor decomposition" and all that. Even dimensions of these objects are non-trivial problem, as far as I understand.
Commutative algebra usually has quite a few accessible papers, and many problems there have strong links to algebraic geometry. So visit math.AC. Good luck!