Exploring areas to move to (Montpellier)

Hello! My husband and I are visiting Montpellier the first week in June to explore it as a possible place for us to relocate to. We want to explore the area and find nice neighborhoods. Do you have recommendation of neighborhoods in Montpellier, or smaller towns outside of Montpellier that are nice? Do you think we would need to rent a car or is public transport sufficient. We also plan on taking day trips...to Toulouse for sure and maybe a couple of other places. Please let us know what you think! Thanks!