We study best-response type learning dynamics for zero-sum polymatrix games under two information settings. The two settings are distinguished by the type of information that each player has about the game and their opponents' strategy. The first setting is the full information case, in which each player knows their own and their opponents' payoff matrices and observes everyone's mixed strategies. The second setting is the minimal information case, where players do not observe their opponents' strategies and are not aware of any payoff matrices (instead they only observe their realized payoffs). For this setting, also known as the radically uncoupled case in the learning in games literature, we study a two-timescale learning dynamics that combine smoothed best-response type updates for strategy estimates with a TD-learning update to estimate a local payoff function. For these dynamics, without additional exploration, we provide polynomial-time finite-sample guarantees for convergence to an $ε$-Nash equilibrium.