What island should we visit?

My boyfriend and I (29 & 27) are looking to visit an island in the Caribbean in May, and are trying to decide which island to visit. We would be there for one week, are not interested in resorts, and are hoping to focus on both beaches and adventuring (such as beginner snorkeling or hiking). We would prefer to have options for good food and city vibes, if possible. Neither of us have traveled to the Caribbean before, so we have little know how about the different islands.

Our budget is flexible but under 4k for both of us (we like to travel cheaply when possible). We live on the east coast, so that is a doable budget for us as well.