Popularly known as "Bica", it's a park downtown with a mini zoo and a playground. The trees are very old, very tall and very impressive! Some (harmless) animals are running free, it's an oasis of peace right in the middle of the city. It's a nice place for a picnic and just relaxing. You'll see birds and monkeys of many kinds, captive or not.
In order to visit the botanical garden (also known as "Mata do Buraquinho"), you'll have to book a guided tour. It's a natural reserve and the amount of visits per day are strictly limited. They have tours in the morning and in the afternoon and you're obliged to wear a pair of jeans and closed shoes (sport shoes will do). It's like a wild version of the Bica, so all the animals there will be running free.
There you'll be able to spot sweeties like marmosets and sloths, foxes, cavies, anteaters, agoutis, countless butterflies and birds, all sorts of ants and spiders and of course snakes. There's no need to be afraid, the local guides are well trained and know exactly what to do in case a snake suddenly appears, it seldom happens anyway.