The Botanical Garden in St Louis, Missouri, is considered to be one of the very best botanical gardens in the world, and a visit to it is a surprisingly interesting way to spend an afternoon.
It has dozens of different gardens, displays and experimental plots, with just about every mundane or exotic plant you could think of. No matter what part of the world you're from, there's a section of the garden to be found that'll make you feel right at home.
There are lots of fountains, including several that people are encouraged to walk through. It also has many highly unusual sculptures and statues, several streams, two or three forests and a lake. For those not interested in plants, there's also a restaurant, a gift shop, and an exhibit of porcelain hummingbirds. There's even a hedge maze.
It's well worthwhile wandering through the garden for a day (or several, since it's so large that it's next to impossible to see all of it at once). This is a good place to go if you're looking for a quiet stroll.
Especially interesting areas of the garden include:
The Koi Bridge - One of the more unusual sites in the garden is the famous Koi Bridge, named after the Koi carp that gather there. It is something of a tradition for visitors to the park to throw morsels of food to the fish, who as a consequence have become so accustomed to being fed that they are virtually tame. Whenever someone passes over the bridge, the carp try desperately to attract the person's attention, which they do by attempting to climb up on top of each other, making peculiar gulping and gasping sounds as if they were drowning. As fish are unable to climb and are perfectly capable of breathing in water, these antics are often quite amusing. The fish are also engaged in a power struggle with the local ducks, who often attempt to steal the discarded food before the carp can reach it.
The Climatron - This is a gigantic geodesic dome1 that serves as a tropical greenhouse. The rainforest plants growing inside are quite beautiful, especially when the orchids are in bloom. This is also the best part of the gardens to visit on a cold day, as it's always pleasantly warm inside.