In this work, we aim to teach robots to manipulate various thin-shell materials. Prior works studying thin-shell object manipulation mostly rely on heuristic policies or learn policies from real-world video demonstrations, and only focus on limited material types and tasks (e.g., cloth unfolding). However, these approaches face significant challenges when extended to a wider variety of thin-shell materials and a diverse range of tasks.On the other hand, while virtual simulations are shown to be effective in diverse robot skill learning and evaluation, prior thin-shell simulation environments only support a subset of thin-shell materials, which also limits their supported range of tasks. To fill in this gap, we introduce ThinShellLab - a fully differentiable simulation platform tailored for robotic interactions with diverse thin-shell materials possessing varying material properties, enabling flexible thin-shell manipulation skill learning and evaluation. Building on top of our developed simulation engine, we design a diverse set of manipulation tasks centered around different thin-shell objects. Our experiments suggest that manipulating thin-shell objects presents several unique challenges: 1) thin-shell manipulation relies heavily on frictional forces due to the objects' co-dimensional nature, 2) the materials being manipulated are highly sensitive to minimal variations in interaction actions, and 3) the constant and frequent alteration in contact pairs makes trajectory optimization methods susceptible to local optima, and neither standard reinforcement learning algorithms nor trajectory optimization methods (either gradient-based or gradient-free) are able to solve the tasks alone. To overcome these challenges, we present an optimization scheme that couples sampling-based trajectory optimization and gradient-based optimization, boosting both learning efficiency and converged performance across various proposed tasks. In addition, the differentiable nature of our platform facilitates a smooth sim-to-real transition. By tuning simulation parameters with a minimal set of real-world data, we demonstrate successful deployment of the learned skills to real-robot settings. ThinShellLab will be publicly available. Video demonstration and more information can be found on the project website https://vis-www.cs.umass.edu/ThinShellLab/.