We present a very high statistics study of D and D_s semileptonic decay form factors on the lattice. We work with MILC N_f=2+1 lattices and use the Highly Improved Staggered Quark action (HISQ) for both the charm and the strange and light valence quarks. We use both scalar and vector currents to determine the form factors f_0(q^2) and f_+(q^2) for a range of D and D_s semileptonic decays, including D to pi and D to K. By using a phased boundary condition we are able to tune accurately to q^2=0 and explore the whole q^2 range allowed by kinematics. We can thus compare the shape in q^2 to that from experiment and extract the CKM matrix element |V_cs|. We show that the form factors are insensitive to the spectator quark: D to K and D_s to eta_s form factors are essentially the same, which is also true for D to pi and D_s to K within 5%. This has important implications when considering the corresponding B/B_s processes.