Knowledge of bubble dynamics provides a microscale or mesoscale basis to further understand the macroscale complex behavior of bubbly flow or extract the constitutive correlations for continuum-based multifluid computational fluid dynamics (CFD) models. This study attempts to investigate the bubble dynamics from a systematic and multiscale perspective, that is, progressively probing the behavior of a single bubble, a bubble pair, and a bubble swarm with a Multiple-Relaxation-Time (MRT) lattice Boltzmann method (LBM). The MRT-LBM method is used to solve the phase-field model and can accurately capture the interface evolution under different flow conditions. This study also incorporates the force term in a different way, and the model is validated by classical numerical tests for a static bubble and capillary wave. Numerical tests show that the model is able to simulate the gas liquid flows with a large kinematic viscosity ratio (up to 1000) and the bubbles of lower Morton number and higher Reynolds number. The simulation results of the terminal rise velocity of air bubbles are comparable with the classical experimental correlations. Interactions of two bubbles vertically or horizontally aligned are simulated, reproducing the trailing bubble catching up with the leading one for a vertically aligned bubble pair and the repeatedly bouncing and attracting phenomenon for a horizontally aligned bubble pair. The behavior of bubble pairs in a bubble swarm is analyzed with the model to understand the different bubble bubble interactions and their influence on micromixing.