Multi-agent reinforcement learning has made substantial empirical progresses in solving games with a large number of players. However, theoretically, the best known sample complexity for finding a Nash equilibrium in general-sum games scales exponentially in the number of players due to the size of the joint action space, and there is a matching exponential lower bound. This paper investigates what learning goals admit better sample complexities in the setting of $m$-player general-sum Markov games with $H$ steps, $S$ states, and $A_i$ actions per player. First, we design algorithms for learning an $ε$-Coarse Correlated Equilibrium (CCE) in $\widetilde{\mathcal{O}}(H^5S\max_{i\le m} A_i / ε^2)$ episodes, and an $ε$-Correlated Equilibrium (CE) in $\widetilde{\mathcal{O}}(H^6S\max_{i\le m} A_i^2 / ε^2)$ episodes. This is the first line of results for learning CCE and CE with sample complexities polynomial in $\max_{i\le m} A_i$. Our algorithm for learning CE integrates an adversarial bandit subroutine which minimizes a weighted swap regret, along with several novel designs in the outer loop. Second, we consider the important special case of Markov Potential Games, and design an algorithm that learns an $ε$-approximate Nash equilibrium within $\widetilde{\mathcal{O}}(S\sum_{i\le m} A_i / ε^3)$ episodes (when only highlighting the dependence on $S$, $A_i$, and $ε$), which only depends linearly in $\sum_{i\le m} A_i$ and significantly improves over existing efficient algorithm in the $ε$ dependence. Overall, our results shed light on what equilibria or structural assumptions on the game may enable sample-efficient learning with many players.