This paper presents two algorithms, AdaOFUL and VARA, for online sequential decision-making in the presence of heavy-tailed rewards with only finite variances. For linear stochastic bandits, we address the issue of heavy-tailed rewards by modifying the adaptive Huber regression and proposing AdaOFUL. AdaOFUL achieves a state-of-the-art regret bound of $\widetilde{O}\big(d\big(\sum_{t=1}^T ν_{t}^2\big)^{1/2}+d\big)$ as if the rewards were uniformly bounded, where $ν_{t}^2$ is the observed conditional variance of the reward at round $t$, $d$ is the feature dimension, and $\widetilde{O}(\cdot)$ hides logarithmic dependence. Building upon AdaOFUL, we propose VARA for linear MDPs, which achieves a tighter variance-aware regret bound of $\widetilde{O}(d\sqrt{HG^*K})$. Here, $H$ is the length of episodes, $K$ is the number of episodes, and $G^*$ is a smaller instance-dependent quantity that can be bounded by other instance-dependent quantities when additional structural conditions on the MDP are satisfied. Our regret bound is superior to the current state-of-the-art bounds in three ways: (1) it depends on a tighter instance-dependent quantity and has optimal dependence on $d$ and $H$, (2) we can obtain further instance-dependent bounds of $G^*$ under additional structural conditions on the MDP, and (3) our regret bound is valid even when rewards have only finite variances, achieving a level of generality unmatched by previous works. Overall, our modified adaptive Huber regression algorithm may serve as a useful building block in the design of algorithms for online problems with heavy-tailed rewards.