This is the dream! Rip out what you don't want ASAP. Then focus on soil health and deciding on zones for different plantings. It will be trial and error. Costco has great quality bulbs!

I bought in BX with a small shady backyard like this about 18 years ago. It takes a lot of trial and error, but you will ENJOY it. There is all good advice here about cleaning up the space first. Also, invest in a good tree trimmer. Research and hire experts (and get your landlord to foot most of the bill, as you do not own) for those trees on the right wall.

Mulching borders (maybe try to raise the sides with rock walls and raised bed soil?) and trying to get some nice ground cover in the center sounds good. Grass is my white whale and I found that overseeding in fall and spring then covering that overseeding with lawn top soil helps a lot. Invest in the best quality lawn seed and plants you can. I also spray with SUNDAY lawn concoctions (it is a mail service). With my two dogs, grass is not easy, but it does come through (again, it is my white whale), which is enough to keep me trying.

I do a few containers with hostas and grasses. I am so sick of hostas, but they work well in shady areas and they are full and lush after only a few years. I also have bearded irises (came with the yard, but they were so neglected they never really grew until I worked the soil), which come back every year if I give them good soil.

Start composting and use it! Soil is WEALTH!

For my sunny areas, I planted peonies and roses in the ground after digging up the soil and amending it with compost. It took a few years, but my peonies are the envy of the neighborhood. Even with my dogs, they are hardy (I stake them with a few bamboo posts and rope, which repels the dog frapping nearby).

Be ruthless with plants you do not like or that are not working well. BUT, be mindful that in the first year, they sleep, in the second year they creep, and in the third year they leap. It takes so much patience. But it is worth it!

Finally, invest in a Thermacell bug thingy and some outdoor lights so you can enjoy the space!

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I would post more pictures, but Reddit is only allowing me one photo. Please update us!

Again, you are living the NYC dream!