The Boboli gardens are directly behind the Palazzo Pitti and are sort-of the major park of the center of Florence, covering eleven acres of land. They are a great place for a leisurely stroll, offering fantastic sweeping views of the Florentine skyline from multiple points within the gardens. On a nice day you can see locals soaking up the sun on some of the various lawns and open spaces scattered throughout. In the fall you can run and jump in the leaves that have fallen from the surrounding trees. The gardens are a great place to come and relax, enjoy a bit of nature, and get away from all the stone and hustle of the city outside the Palazzo gates. The gardens extend from the Fort of the Belvedere and the adjacent Bardini Gardens to the north (in fact some of the garden walls are shared with the fort) and the Porta Romana (one of the old doors through the original city walls) on the south end. 
The gardens once belonged to the Medici family, when they lived in the Palazzo Pitti as Grand Dukes of Tuscany. They are formal, 16th-century gardens, which is a style that focuses on wide gravel avenues, long stretches of grass, greenery and statuary. In keeping with the Medici tradition of being patrons of the arts, the garden is home to a number of important statues and fountains that represent the height of 16th century sculpture. Directly behind the Palazzo, and heading up the hill, is the Fountain of Neptune (different from the one outside Palazzo Vecchio). Also here is a small pond with a sculptured island in it, called Island Fountain, further down the axis of the gardens. There are a number of other sculptures and fountains to run into and discover. The Mostaccini fountain is particularly delightful, and the famous statue of a naked and fat Bacchus riding on a turtle is also here (just by the main exit of the gardens back into the courtyard of Palazzo Pitti). If traditional statues are not your favorite, the gardens also have a few modern sculptures in place all around the grounds. It is fun to just wander around and bump into a new statue at every turn. 
There are not many flowers in the Boboli Gardens, as that would not be in keeping with the 16th-century style, but some flower beds can be found outside the Porcelain Museum (which is inside the garden grounds). The Porcelain Museums houses a collection of a number of different fine serving pieces owned by the Medici royal family. Also of interest in the grounds are the Medici grottoes, which are a sort of artificial cave decorated with face stalactites and artworks. There are two grottoes, both nearby the exit to the Palazzo Courtyard. The larger of the two, the Buontalenti Grotto, is down the same pathway as the Bacchus fountain. The other grotto, the Grotto of Madama, is a little bit more off the beaten path but maps of the gardens will locate it for you. Both of the grottoes were started by famous artist and art historian Giorgio Vasari. You cannot go inside the grottoes, but they are certainly worth a look as they are very different from anything else within the gardens. 
A ticket can be purchased at the main ticket desk and will cost 10€ for the combined ticket number two (which also includes the Silver Museum, the Bardini Gardens, the Costume Gallery and the Porcelain Museum), but entry is free to those that hold the Friends of the Uffizi card. After a day at the gardens and in the Palace, stop by Pitti Gola and Cantina, a small wine bar across the square from the Palazzo that serves excellent wine in addition to small plates of antipasti such as cheese and bruschetta. Also nearby, a few blocks up the street, is the famous Ponte Vecchio. A stroll across the bridge, looking at all the glittering jewelry for sale there and getting a good look at both sides of the Arno River, is worth the time.
