Matt and Mary Caroline spend just about every Saturday and many days throughout the summer fishing, surfing, and hunting for sharks' teeth at Guana Beach in South Ponte Vedra. It has three public access points with parking lots making it convenient and less busy than some of the other nearby beaches. Be sure to bring $3 in cash to pay for a parking ticket for your windshield!
St. Johns Town Center is an outdoor mall that offers many shops (Nordstrom, Dillards, Dick's Sporting Goods, Barnes and Noble, etc) as well as restaurants for a fun day of shopping!
Avondale is another shopping area that has a more boutiquey feel with more local shops/boutiques and restaurants compared to the Town Center.
Jacksonville has a wonderful zoo if you are looking for an outdoor activity that isn't the beach!
The Cummer has wonderful indoor art exhibits with beautiful gardens outside on the river.
The Jacksonville Beach Pier is another great option for public beach access. It is typically busier than Guana, but it also has many restaurants and bars nearby to enjoy as well!