You would need to take a water taxi to get to Peanut Island. But it is a nice little water taxi ride from Sailfish Marina, which is a nice place to grab a bite to eat and feed the fish.

Personally, a ride south along the water in Palm Beach, with a trip down Lake Ave in Lake Worth for restaurants/bars or all the way to Atlantic Ave in Delray Beach.

If you want a ride up the coast, the ride along A1A has some nice spots starting at the Blue Heron Bridge (where you would need to go to get to the above mentioned Sailfish Marina). you can go along all the way up to Peck Lake State Park in Hobe Sound, passing through Jupiter Island, where you can say hi to the likes of Tiger Woods and Alan Jackson. The beach at Jupiter is amazing, and the views from either Jupiter Inlet Park, Dubois Park, or Jupiter Lighthouse park.
Along the way, there are several places to eat, drink and be merry.
I would recommend Guanabana's for brunch/lunch, Square Grouper (of Alan Jackson's 5 o'clock somewhere fame) for drinks and a spectacular view of the inlet and lighthouse, or Jetty's for dinner.

If you are in Palm Beach on Thursday, I would recommend spending the evening across the intracoastal waterway on Clematis Street for Clematis by Night, where they have music, food and vendors.