There are so many unique places to visit in Miami that have nothing to do with the beach.
If you're interested in experiencing authentic Cuban culture, check out Calle Ocho, where you can see domino games in action, eat authentic Cuban food at ventanitas, shop at local stores, and take in art.
Since you probably have a bathing suit in tow, you can drive to the historic Venetian Pool in Coral Gables. It has waterfalls and grottos, and I've never seen a pool quite like it.
If you're into outdoor adventures, you can visit parks that offer land-based activities, like biking and hiking, and water adventures, like kayaking and swimming.