First and foremost, you have to see the beaches on the island. The Sandbar is a great place to start because its one of the only restaurants in the sand. Its got great drinks and an amazing view of the gulf sunset. Another beach to consider would be Bean Point, which is at the northern most tip of the island. Its where the Gulf meets the Bay, it typically offers more privacy than the other beaches. Or if you're staying further south on the island Bradenton Beach is another popular beach area to go to.
Pine Avenue has lots of shops and restaurants to cool off in for when you're burnt out of the beach. There's a custom donut shop called Donut Experiment. There's a great Mexican restaurant called Poppo's Taqueria. The boutiques aren't the normal souvenir shops where you buy gimmicky shirts and seashells either. Places like Pink & Navy and PIneapple Junktion Co-op have cute stuff that wont make you feel like a typical tourist.