The farther you travel from the center of Vero Beach, the lighter the crowds. Round Island Park has both oceanside and riverside components. The beachside park is favored by surfers, so you may want to avoid that hectic scene or just hang out on the playground.
Across the road on the Indian River, you have the opportunity to walk a trail and maybe spot manatees. Outfitters will deliver rental kayaks to launch into the calm, mangrove-lined waters.