This paper proposes FAST-Calib, a fast and user-friendly LiDAR-camera extrinsic calibration tool based on a custom-made 3D target. FAST-Calib supports both mechanical and solid-state LiDARs by leveraging an efficient and reliable edge extraction algorithm that is agnostic to LiDAR scan patterns. It also compensates for edge dilation artifacts caused by LiDAR spot spread through ellipse fitting, and supports joint optimization across multiple scenes. We validate FAST-Calib on three LiDAR models (Ouster, Avia, and Mid360), each paired with a wide-angle camera. Experimental results demonstrate superior accuracy and robustness compared to existing methods. With point-to-point registration errors consistently below 6.5mm and total processing time under 0.7s, FAST-Calib provides an efficient, accurate, and target-based automatic calibration pipeline. We have open-sourced our code and dataset on GitHub to benefit the robotics community.