Heart of coconut grove, a cute town to walk around!
Another great shopping area with a bunch of restaurants, coffee shops, etc.
Key Biscayne Beach isn't too far (closer than South Beach) if you want some extra sunshine! Hobbie Island Beach Park is immediately after the Rickenbacker Bridge. Across the street from Hobby Island Beach Park are a few great food & drink spots: Gramps for casual drinks and food, and Rusty Pelican for a more formal sit-down meal. If you venture further out on Key Biscayne, you can visit Brandon Park Beach or drive all the way to the end to enter Bill Baggs Cape Florida State Park and the Lighthouse beach. One of our favorite drink spots is over here: The Cleat :)
If you want to head to South Beach, it's approximately 30 minutes (without traffic) from the Grove/Gables areas. There are plenty of public beach areas and restaurants to explore!