Learning from repeated play in a fixed two-player zero-sum game is a classic problem in game theory and online learning. We consider a variant of this problem where the game payoff matrix changes over time, possibly in an adversarial manner. We first present three performance measures to guide the algorithmic design for this problem: 1) the well-studied \emph{individual regret}, 2) an extension of \emph{duality gap}, and 3) a new measure called \emph{dynamic Nash Equilibrium regret}, which quantifies the cumulative difference between the player's payoff and the minimax game value. Next, we develop a single parameter-free algorithm that \emph{simultaneously} enjoys favorable guarantees under all these three performance measures. These guarantees are adaptive to different non-stationarity measures of the payoff matrices and, importantly, recover the best known results when the payoff matrix is fixed. Our algorithm is based on a two-layer structure with a meta-algorithm learning over a group of black-box base-learners satisfying a certain property, along with several novel ingredients specifically designed for the time-varying game setting. Empirical results further validate the effectiveness of our algorithm.