I love to enjoy the flowers and plants. It is a lot of hard work and patience to grow them, and I guess that is why the gardening can be called nursery.
Living in NYC, there is no backyard of any sort, but if you are willing to walk just a little there are flowers and gardens to view, especially in the spring time, and it is all free! Thanks to the diligent professional gardeners and volunteers.
There is one garden off west end Ave and 92nd street. Have you seen the movie "You've got mail". In the end of that movie, Meg Ryan and Tom Hanks finally had their first kiss in this very garden knowing they were the people fallen in love with for a long time. It is a mini garden and totally maintained by volunteers' work.
There is central park of course. Central park itself is a big beautiful garden.
I also often visited the Brooklyn botanical garden and Bronx botanical garden. Both are readily accessible by subways. As I remember both are free in Saturday mornings for admission. Brooklyn botanical garden has a very strong oriental taste. As a matter of fact the garden has a Japanese garden within. Bronx botanical garden is somewhat more grandiose, and tidy. The peony thrived there!
There is another best kept secret in NYC. At the very north end of Mahattan, there is very nice neighborhood north of 190th street. (This place is just a few steps away to the famous Columbia University medical school, which itself is in a rather poor neighborhood). Surrounding The Cloisters Museum, the view is stunting with nice plantations, beautiful cloisters, GW bridge, Hudson river, and even some chiseled cliffs right over at the NJ side.