To achieve sample efficiency in reinforcement learning (RL), it necessitates efficiently exploring the underlying environment. Under the offline setting, addressing the exploration challenge lies in collecting an offline dataset with sufficient coverage. Motivated by such a challenge, we study the reward-free RL problem, where an agent aims to thoroughly explore the environment without any pre-specified reward function. Then, given any extrinsic reward, the agent computes the policy via a planning algorithm with offline data collected in the exploration phase. Moreover, we tackle this problem under the context of function approximation, leveraging powerful function approximators. Specifically, we propose to explore via an optimistic variant of the value-iteration algorithm incorporating kernel and neural function approximations, where we adopt the associated exploration bonus as the exploration reward. Moreover, we design exploration and planning algorithms for both single-agent MDPs and zero-sum Markov games and prove that our methods can achieve $\widetilde{\mathcal{O}}(1 /\varepsilon^2)$ sample complexity for generating a $\varepsilon$-suboptimal policy or $\varepsilon$-approximate Nash equilibrium when given an arbitrary extrinsic reward. To the best of our knowledge, we establish the first provably efficient reward-free RL algorithm with kernel and neural function approximators.