Brooklyn Botanical Garden, Grand Army Plaza
Buy a beef patty and a coco bread from Christie's and eat it in Prospect Park.
Go to one of the farmer's market and have a picnic in Prospect Park or Fort Greene Park (depending on the farmer's market you go to)
Go to the Brooklyn flea market and have brunch afterwards around Fort Greene
When it's nice weather, go to Brooklyn Bridge Park to look out onto the river and Manhattan. Plenty of places to eat. Great to bring a bike too. Definitely a must.