If the pool is your main priority, then focus on getting that in. If you want a pool like your neighbor's, it will probably take up a major portion of that size of yard. So, keep it simple, don't worry about spending money in ways that you don't need it (testing soil, renting tillers, etc.). Google installing inflatable pools---I'm guessing primary considerations are getting a flat, smooth surface area, maybe covering it with a layer of sand (all of which you can do with a shovel and elbow grease), setting it as far out from under trees as possible, trimming back the hedges and trees (to give you more sunlight at the pool and reducing the amount of leaves going into your pool), etc. You'll probably also find lots of information about how to fill in around it (mulch, grass seed/bedding, turf, etc.). Once you've got your pool set up, you can see what sort of spaces you have around it and fill in accordingly----create your patio space, maybe put some pavers down for a walkway, a little bench at the back with some potted plants around it, a trellis or two along the perimeter with some flowering vines, etc. If you decide you want to do some vegetable gardening, look into container gardens so you can start with good, clean soil; they also look very nice lined up in the sun along a perimeter. Lastly, even though you are renting and do not own the property, the experience of creating and taking care of such a space can be a very enjoyable and worthwhile hobby. Even if you eventually pass it on to someone else, it's always fun to transform a space into something beautiful. Go for it! Good luck and remember that you can keep this simple. Have fun!
If you are renting, I will say to you the same thing that I say to renters: Don't bother. Why should you invest a lot of blood, toil, sweat, and yes, tears, in improving a piece of property that you will inevitably have to leave behind? It can be a heartbreaker to have to abandon a garden that you've lavished all your love on to the tender mercies of the tenants who come after you, who most likely will not maintain your garden, and will simply mow it all down. Especially because, overall, it's probably going to cost more than $500. The soil testing alone (see below) is usually a couple hundred bucks. And then there's tiller rental, unless you buy one...And soil amendments like compost and composted manure, bone meal, etc. So. Assuming that your landlord and his daughter are on board with this. The first rule of backyard makeovers is that the infrastructure goes in first: patios, pergolas, gazebos, grape arbors, storage sheds, potting sheds, greenhouses, firepits, water features, koi ponds, sidewalks, driveways, flagstone pathways, swimming pools, dog runs, et cetera. This is because if you do the lawn first, you end up ripping out half of it in order to build your patio/pergola/gazebo et cetera. It is killing to have to see it go, after all the blood, sweat, toil, and yes, tears, you invested in nurturing it along for a year (which is, yes, how long it takes to get a good lawn established). Infrastructure also has a way of taking up more room than you had visualized. A 10x20 patio on paper is one thing, but a 10x20 patio, especially in a tiny yard, can take over the whole yard, visually. Your yard can end up looking like nothing BUT patio, or pergola, or gazebo, or koi pond, or whatever. So figure out what you're going to want, get it built, and then turn your attention to the yard. In the meantime, you can be improving your soil. You start by getting some serious soil tests done. Urban backyard soils can be loaded with an array of industrial contaminants, such as lead, heavy metals like cadmium, chemicals like dioxin and PCBs. This is because in bygone eras, industries simply dumped their surplus chemicals, waste products, and slag out back, or in the vacant lot next door to the factory. Then time went by, the factory was torn down, and houses were built on top of the slag. Even if you live in a place that has always been residential, the contaminants can still be airborne, all those generations of smog precipitating out of the air every time it rains, so it's wise to get the soil tested. This is testing that is up one level from the simple tests for NPK and pH that you can buy the kits for. You send your soil away to a serious soil testing lab, and you get your results later. These are unfortunately not cheap, but if you're ever going to want to grow edibles, you really need to know what's in your soil. Digging up and replacing the soil isn't an option unless you have Superfund money. So while you're waiting for your soil test results, you can be planting some green manure crops to begin improving your soil. This is going to require a tiller. If you own one or can borrow one on a regular basis, that's awesome, because otherwise you have to rent them, and it's not something you can transport on the subway. I have a probably skewed vision of all New Yorkers not owning cars and using the subways and taxis to get around. Anyway, a green manure is something that you plant that grows fast, and 8 weeks later you till it in. It breaks down and adds organic matter to the soil, thus improving it. A tiller is also used for the weed control protocol, which you can also do while you're waiting, is where you till it all up, water it, wait for the buried weed seeds you just brought to the surface to sprout, then you till them under, cackling viciously. Since this brings even more weed seeds to the surface, you water it again, wait, and you till those under when they sprout. You can keep this up--till, water, wait, till--until, with a faint sense of disappointment, you no longer see green. That means that you've cleared your future lawn of 99% of potential weeds for years to come. Last but not least, contact your local university extension office. It's their federally mandated, taxpayer-funded job to help homeowners with garden problems. They will have planting guides, charts, brochures, and tons of free advice. Google "[city or county] university extension office master gardener" and look around for a phone number or email.