Florence abounds with historic sites, world-class art, and of course many tourists. What it doesn't have as much of, especially in its historic heart, is green spaces with the notable exception of the Boboli Gardens of the Pitti Palace. Even there, however, you may find that you are sharing the manicured gardens with hundreds of other tourists.
All this means the respite offered by Florence's Botanical Gardens (or Orto Botanico, in Italian) is especially welcome. Keep walking one block beyond the Accademia and you'll find Europe's third oldest botanical garden, established in 1545 by Cosimo I de'Medici. (It's almost contemporary with the oldest and second oldest, Padua and Pisa. Padua was also established in 1545; Pisa in 1544, though it moved to its current location in 1591).
The garden is small, at under six acres, but home to some 5,000 specimens. The greenhouses date from the mid-19th century. Of special note are several trees that are centuries old as well as a large collection of azaleas, which draws visitors when they are in bloom (April to May). The gardens are open only from April 1 to October 15, daily from 10 am to 7 pm.