Manhattan has Park Avenue, Brooklyn has Park Slope! Don't miss this beautiful historical residential area of Brooklyn, located along the Flatbush Avenue and Prospect Park West, passing by the Grand Army Plaza. This stylish area of Brooklyn became home for a Brooklyn elite in 1800's.
Today Park Slope is the largest and the most beautiful (and expensive) residential area of Brooklyn.
Besides the beautiful houses, there are a lot of places for shopping, dining, and entertainment.
Park Slope also hosts the Brooklyn Botanical Gardens and the Brooklyn Museum -- both well-worth a visit. South of Park Slope is the neighborhood Sunset Park where Greenwood Cemetery is located. It is a beautiful cemetery with a lot of history. It would make for a lovely walk in the spring, summer or fall (though be aware that bicycle riding is prohibited).
7th Avenue in Park Slope hosts the areas primary business district. However, 5th Avenue is quickly catching up to its up-hill rival. 5th Avenue hosts a variety of new restaurants and bars, antique shops, and small boutiques.
I wouldn't call Brooklyn quaint -- even in some of the best areas. However, it offers an interesting glimpse into a very different kind of New York than Manhattan. You will walk in and out of immigrant neighborhoods and long-established ethnic neighborhoods as well as wealthy and working-class neighborhoods. You will find beautiful turn of the century brownstone housing and a few Civil War-era single-family houses.
Other areas of interest are Brooklyn Heights (has always been a more tonier neighborhood with a beautiful promenade along the water that has a view of lower Manhattan); Cobble Hill where you will find elegant brownstones; Carroll Gardens (an old Italian neighborhood that has recently experienced a boom of restaurants and small shops on its now famous Smith Street). If you are willing to explore further into Brooklyn, you might want to check out the Weeksville houses in Bedford-Stuyvesant. Weeksville consists of Civil War era houses that housed free African Americans.