Fair clustering is a constrained variant of clustering where the goal is to partition a set of colored points, such that the fraction of points of any color in every cluster is more or less equal to the fraction of points of this color in the dataset. This variant was recently introduced by Chierichetti et al. [NeurIPS, 2017] in a seminal work and became widely popular in the clustering literature. In this paper, we propose a new construction of coresets for fair clustering based on random sampling. The new construction allows us to obtain the first coreset for fair clustering in general metric spaces. For Euclidean spaces, we obtain the first coreset whose size does not depend exponentially on the dimension. Our coreset results solve open questions proposed by Schmidt et al. [WAOA, 2019] and Huang et al. [NeurIPS, 2019]. The new coreset construction helps to design several new approximation and streaming algorithms. In particular, we obtain the first true constant-approximation algorithm for metric fair clustering, whose running time is fixed-parameter tractable (FPT). In the Euclidean case, we derive the first $(1+ε)$-approximation algorithm for fair clustering whose time complexity is near-linear and does not depend exponentially on the dimension of the space. Besides, our coreset construction scheme is fairly general and gives rise to coresets for a wide range of constrained clustering problems. This leads to improved constant-approximations for these problems in general metrics and near-linear time $(1+ε)$-approximations in the Euclidean metric.