"Jdeme na zahradu." normally means we are going to some particular garden and it is clear which one (normally it is our garden or garden of the house we are now in).
No, it is always na zahradu. Rarely, in specific circumstances, there is "do zahrady".
But zahrada is not a general park, it is someone's garden, often in the backyard.
Some parks in Prague do have zahrada in their name but they used to be private aristocratic gardens as well.