This has potential! I don't know where you are, so I'm not sure exactly what plants will grow well in your yard. But I'd say, create a curved patio with pavers (brick?) and line it with colorful container plantings in strategies places to create the sense of an outdoor room. A water feature of some sort (a fountain, a small wall feature that recycles the water that falls down it) will act as something of a sound barrier by covering up noise from around you. You could then either surround yourself with lattice-type wood fencing through which you've trained vines like clematis or simply plant tall-growing plants against the existing buildings on trellises. Beds that line the space but curve around will be helpful in creating a sense of privacy and retreat. Here are some ideas for urban garden spaces (I like the very first one and the one with the self-watering planter wall, though it sounds pricey). Here's a slideshow of plants for privacy; I'd opt for the climbing plants and bushy shrubs that are more wild looking rather than the formal-looking arborvitae. But..a water feature (check out these basin fountains or the small slate waterfall) will really add a lot.

Along the fence to right (looking outward from the house) you've a row of small trees that extend the screening height of the fence. This is one solution for adding privacy. I'd say the downfall of this particular application is that the tree canopies weren't sheered along the way, so they are loose and wide ranging instead of being dense and solid. Depending on the tree species, they are probably reclaimable but in order to do that there would need to be some trimming and regrowing, and probably a modest period of less screening in order to get to a better result. The "moral" of the story is keep up with pruning along the way.

Another solution for tall screening is a hedge of skinny "trees" like Arborvitae. You wouldn't need tall screening necessarily where someone's garage limits the view. From these pictures we can't tell how much shade the tall tree at the back produces. In order for plants to grow well below it, light would need to reach them, especially if Arborvitae. You might need to look for shade tolerant narrow tall shrubs instead.

You can use a small tree but the one I see does not have a round canopy. You only need to block the view from the windows. Not much you can do about sound though.

Do you have a budget range in mind or are you in the idea-gathering stage? A good temporary way to block the apartment view could be to install a shade sail on new tall wood posts. Check with city hall for fence height regulations- info is often online.

Do the wood fences belong to you or your neighbors?

There is actually quite a lot you can do for sound. Water features, especially those that splash, are excellent design devices for sound suppression.

Unless your photos are not providing the full picture, it looks like privacy issues are more or less already successfully addressed by existing plantings. The exception is the corner area next to the house where the narrow existing tree is located. And some judicious pruning could theoretically help with that to broaden the canopy (depending on the species involved). Although the plants are not evergreen, most backyard usage is during the warmer summer months when the trees are in leaf and will provide a privacy screen. You can also increase privacy by some sort of structure placed where it will block incoming views/sightlines.

But how all this should be compiled and arranged really depends on the development of a decent design plan. I would consider contracting with a local designer to help you with this and to maximize a sense of privacy and usage of this area.

Just as an aside, but I always wonder why foks are reluctant to hire a local professional to help with these issues and prefer to resort to an online advice forum. Even more so if they are not experienced gardeners or feel hampered by their ability to develop an effective layout and planting plan. This is what garden designers do :-)) And very efficiently and with a very modest cost.

Wind chimes if you don't hate them, also mitigate noise pollution.

Thank you all for your great advice! Of course we will be hiring a professional in the field once we get closer but it is helpful to get some ideas to start from. We're in Northern CA so luckily plants grow well and almost year round. Although I love the idea of a water feature they tend to attract rodents in our area which is near a bay. And chimes might drive my neighbors nuts. They are super close.

Funny, my neighbor in my row house had them and I quite liked them. YMMV of course.

Edited to add that there are actually trees that rustle and create a pleasant sound. The classic examples are members of the poplar family. They can get weedy but can also be beautiful if you don't mind some time spent on management. Silver maples and birches also have this feature to some extent. Not sure about the examples from your zone, but plants adapted to windy slopes can sometimes provide this element.