We present a systematic study of the structural and magnetic properties of two branches of the rare-earth tripod-kagome-lattice (TKL) family A(2)R(3)Sb(3)O(14) (A= Mg, Zn; R= Pr, Nd, Gd, Tb, Dy, Ho, Er, Yb; here, we use abbreviation A-R, as in MgPr for Mg2Pr3Sb3O14), which complements our previously reported work on MgDy, MgGd, and MgEr [Z. L. Dun et al., Phys. Rev. Lett. 116, 157201 (2016)]. The present susceptibility (chi(dc), chi(ac)) and specific-heat measurements reveal various magnetic ground states, including the nonmagnetic singlet state for MgPr, ZnPr; long-range orderings (LROs) for MgGd, ZnGd, MgNd, ZnNd, and MgYb; a long-range magnetic charge ordered state for MgDy, ZnDy, and potentially for MgHo; possible spin-glass states for ZnEr, ZnHo; the absence of spin ordering down to 80 mK for MgEr, MgTb, ZnTb, and ZnYb compounds. The ground states observed here bear both similarities as well as striking differences from the states found in the parent pyrochlore systems. In particular, while the TKLs display a greater tendency towards LRO, the lack of LRO in MgHo, MgTb, and ZnTb can be viewed from the standpoint of a balance among spin-spin interactions, anisotropies, and non-Kramers nature of single-ion state. While substituting Zn for Mg changes the chemical pressure, and subtly modifies the interaction energies for compounds with larger R ions, this substitution introduces structural disorder and modifies the ground states for compounds with smaller R ions (Ho, Er, Yb).