Soft materials, with the sensitivity to various external stimuli, exhibit high flexibility and stretchability. Accurate prediction of their mechanical behaviors requires advanced hyperelastic constitutive models incorporating multiple parameters. However, identifying multiple parameters under complex deformations remains a challenge, especially with limited observed data. In this study, we develop a physics-informed neural network (PINN) framework to identify material parameters and predict mechanical fields, focusing on compressible Neo-Hookean materials and hydrogels. To improve accuracy, we utilize scaling techniques to normalize network outputs and material parameters. This framework effectively solves forward and inverse problems, extrapolating continuous mechanical fields from sparse boundary data and identifying unknown mechanical properties. We explore different approaches for imposing boundary conditions (BCs) to assess their impacts on accuracy. To enhance efficiency and generalization, we propose a transfer learning enhanced PINN (TL-PINN), allowing pre-trained networks to quickly adapt to new scenarios. The TL-PINN significantly reduces computational costs while maintaining accuracy. This work holds promise in addressing practical challenges in soft material science, and provides insights into soft material mechanics with state-of-the-art experimental methods.