Phononic bandgap metamaterials, which consist of periodic cellular structures, are capable of absorbing energy within a certain frequency range. Designing metamaterials that trap waves across a wide wave frequency range is still a challenging task. In this study, we proposed a deep feature learning-based framework to design cellular metamaterial structures considering two design objectives: bandgap width and stiffness. A Gaussian mixture variational autoencoder (GM-VAE) is employed to extract structural features and a Gaussian Process (GP) model is employed to enable property-driven structure optimization. By comparing the GM-VAE and a regular variational autoencoder (VAE), we demonstrate that (i) GM-VAE has the advantage of learning capability, and (ii) GM-VAE discovers a more diversified design set (in terms of the distribution in the performance space) in the unsupervised learning-based generative design. Two supervised learning strategies, building independent single-response GP models for each output and building an all-in-one multi-response GP model for all outputs, are employed and compared to establish the relationship between the latent features and the properties of interest. Multi-objective design optimization is conducted to obtain the Pareto frontier with respect to bandgap width and stiffness. The effectiveness of the proposed design framework is validated by comparing the performances of newly discovered designs with existing designs. The caveats to designing phonic bandgap metamaterials are summarized.