  This is the first part of a series of three articles. In this paper, we
obtain weighted norm inequalities for different conical square functions
associated with the Heat and the Poisson semigroups generated by a second order
divergence form elliptic operator with bounded complex coefficients. We find
classes of Muckenhoupt weights where the square functions are comparable and/or
bounded. These classes are natural from the point of view of the ranges where
the unweighted estimates hold. In doing that, we obtain sharp weighted change
of angle formulas which allow us to compare conical square functions with
different cone apertures in weighted Lebesgue spaces. A key ingredient in our
proofs is a generalization of the Carleson measure condition which is more
natural when estimating the square functions below $p=2$.
