Fix d≥3 and 1<p<∞. Let V:Rd→[0,∞) belong to the reverse Hölder class RHd/2 and consider the Schrödinger operator LV:=−Δ+V. In this article, we introduce classes of weights w for which the Riesz transforms ∇LV−1/2, their adjoints LV−1/2∇ and the heat maximal operator supt>0e−tLV|f| are bounded on the weighted Lebesgue space Lp(w). The boundedness of the LV-Riesz potentials LV−α/2 from Lp(w) to Lν(wν/p) for 0<α≤2, 1<p<dα and 1ν=1p−αd will also be proved. These weight classes are strictly larger than a class previously introduced by Bongioanni, Harboure and Salinas in [8] that shares these properties and they contain weights of exponential growth and decay.The classes will also be considered in relation to different generalised forms of Schrödinger operator. In particular, the Schrödinger operator with measure potential −Δ+μ, the uniformly elliptic operator with potential −divA∇+V and the magnetic Schrödinger operator (∇−ia)⁎(∇−ia)+V will all be considered.Finally, this article will investigate necessary conditions that a weight w must satisfy in order for the Riesz transforms or the heat maximal operator to be bounded on Lp(w). To aid in this task, lower bounds for the heat kernel of the standard Schrödinger operator −Δ+V will be proved. These estimates provide a lower counterpart to the upper estimates proved in [23].