Miami Beach has a lot of great restaurants or spots to relax and soak up the sun. Traveling to north beach could take a while with traffic, but south beach is only 15 minutes away. If you would like to avoid the touristy part of the beach, head south all the way to the tip of the island.
Star Island is located between downtown and Miami Beach and contains many lavish mansions, which are often owned by celebrities. While you cannot drive onto star island without permission, there are boat tours that leave out of downtown and cruise around the island while identifying celebrity homes.
The beaches on Key Biscayne are about 15 minutes away from downtown and are isolated, so you will avoid tourism by going here. They are usually pretty quiet and always beautiful!