Might feel less sciencey than a telescope, but a lot of the things you're talking about are best seen through binoculars.

I'm not particularly into amateur astronomy, but I have an ex who was really into it, so I learned a lot about the gear that people like. And I've taught classes in optics, so I've picked up a few things by osmosis.

As far as name-brands go, everyone I know who's into this stuff loves their Celestron equipment. Meade is the old standard, but I've heard they've been a little less on their game lately.

The moon is a great thing to start with, and I know people whose favorite observation target is still the moon after many years. But it's a little tricky as far as telescopes go. Most telescopes are made for deep sky objects, but the moon is A LOT brighter than deep sky objects. So usually if you want to look at the moon through a telescope you need to use a "moon filter," basically a sunglass lens that you put behind the eyepiece. It's no big deal; like a $50 add-on. But keep it in mind if you decide on the telescope route. Binoculars collect a lot less light, so you can usually look comfortably at the moon through binoculars. And good astro binoculars have enough magnification to make out individual craters and even see shadows on the craters, so you can really see the 3-d structure of it in a very satisfying way.

One issue with binoculars is that if you're hand-holding them, the image will jitter. It's not a big deal if you're looking at the moon, but for objects farther away it can get to be a big deal. One fix is active motion correction, but that tends to raise the price to about $1k, so more than you'd want to spend at this stage. The nice thing is, good astro binoculars (and most good binoculars, in general) can be mounted on a tripod, which is more than affordable given the dollars you'd save going with binocs instead of a full telescope.

Here, for example, are some that I'd strongly consider if I was thinking about getting into this:

https://www.celestron.com/products/skymaster-pro-20x80-binoculars

Andromeda: little-known fact, but Andromeda is HUGE. The number I remember (which could be wrong, but it's in the ballpark) is that the full size of Andromeda is six times bigger than the moon in the sky from Earth. The core of Andromeda, which is easiest to see, is roughly as big as the moon. So you don't need much magnification to see Andromeda. What you need is to collect a lot of light (big as it is, it's still pretty dim from the Earth). Some good astro binoculars like the ones I linked above would probably be up to the task. Actually, if you're into photography, you could capture it in a regular camera with a long-exposure shot. I've done it on a consumer-grade Canon DSLR without any special mounts or lenses.

That said, light pollution is your enemy when it comes to any deep-sky objects. If you live near NYC, you'd probably need to drive at least an hour out of town to see it over the noise, no matter how good your telescope is. I live in the middle of nowhere, and on a clear night, if I know where to look (and know to avert my eyes because peripheral vision is usually better for dim objects), I can see Andromeda with the naked eye. Then again, where I live, on a clear night I can make out the Milky Way with the naked eye as well. If you can't make out the Milky Way, you probably can't make out Andromeda, no matter how good your telescope is, because the issue is background noise, not magnification or light collection.

Saturn and Jupiter: Those are also great objects, and probably visible from NYC on a clear night. This might be where you want a telescope, though. Planets are pretty small in the sky, even with good binoculars. With binoculars, you might be able to make out that they're disks instead of point-sources, which is cooler than it sounds. And you might be able to see the moons, which is a lot cooler than it sounds, especially if you can watch their motion night after night. But you probably wouldn't make out the rings of Saturn or the bands of Jupiter through binoculars. You can make them out easily with a moderately good telescope.

The standard machine for planet viewing is either a Schmidt-Cassegrain telescope (quite expensive):

https://www.celestron.com/products/edgehd-8-optical-tube-assembly-cg-5-dovetail

or a Dobsonian telescope (quite a bit more reasonable):

https://www.bhphotovideo.com/c/product/1141702-REG/sky_watcher_s11610_8_traditional_dobsonian.html/

BTW, I have friends who have SkyWatcher Dobsonians, and they love them. Keep in mind the "aperture." That is, the diameter of the main mirror. The amount of light you collect is proportional to the surface area of the mirror (pi r^2). So an 8" aperture collects almost twice as much light as a 6". And a 10" collects about 50% more light than an 8". So those two-inch increases in mirror size mean a lot. Usually beginning amateur astronomers start with an 8" Dobsonian. And if they get serious about it they'll upgrade to 10" or even 12" after a few years. 6" might work for some of the things you're interested in, but there are better options at that price point (see below).

The issue with Dobsonians: they're a lot simpler and cheaper than other telescope designs (like the Schmidt-Cas design I mentioned above). But the downside of this simplicity is that you have to "collimate" them yourself every time you set up. Collimation means aligning the various mirrors. This isn't a big deal if you've done it once or twice, but you might want to buy a laser collimator. This is a little thing (basically a laser pointer) that you stick into the eyepiece socket, and then you adjust the mirrors to put the laser spots at the right place. So if you went the Dobsonian route, you'd almost certainly want a moon filter and a laser collimator, which might raise the $415 that I mentioned above right up to your price point of $500. Also, if you're thinking of moving it around, you'd want to make sure the thing fits in your car. An 8" probably would, but you'd want to look up the dimensions and do some measurements to be sure. They're big bulky things, especially the stands.

"What about those motors that'll automatically find things in the sky?" You didn't ask that question, but I'd bet it's come up. Yes: such motors exist, and they're super convenient if you're really into this. But they also raise the price by a lot. They also usually require some significant alignment to set up in the first place. Also, by finding sky objects automatically, they kinda take away half the fun of amateur astronomy. Most people, as they're getting into this stuff, can quickly learn the constellations, and learn how to find interesting objects relative to those constellations by "star hopping." I've heard multiple people say that if the telescope finds objects automatically, it's not much more satisfying than just looking up objects on Wikipedia. So I'd suggest putting those fancy mounts out of your mind for now.

One last point, and maybe the most important: To really get much out of this, you kind of want to know what you're doing, and that's not easy. So usually the best advice for anyone thinking about getting a telescope is: "Don't get a telescope." Instead, find a local astronomy club, join it, attend some meetings and some star parties. They exist pretty much everywhere, they're staffed by old nerds who love this stuff and love explaining it even more. Also: they probably have some really great telescopes that you can either rent or borrow for free! And they can show you how to use those scopes. Just google "NYC astronomy clubs" and find something near where you are. There's a chance there might not be as many in NYC as elsewhere, but I know they exist in LA; and LA has light pollution just as bad as NYC. They can also be a great way to get your kids interested in science, generally speaking. They usually hold a monthly meeting with a talk by an active researcher (I've given dozens of talks about Black Hole physics to local astro clubs, and met a bunch of enthusiastic middle-schoolers). And then they'll usually have a "star party" once or twice a month, on clear nights, where a bunch of people bring their scopes and find things for anyone who comes by. Personally, I'm the kinda guy who prefers just buying my own thing and figuring out how to use it, but having seen enough of the culture of these astronomy clubs I know how valuable they are. My parents got me a cheap telescope when I was a kid. I couldn't find anything, decided it was boring, and never used it again. On the other hand, people who join astronomy clubs (as kids or adults) can meet people to answer all of their questions, show them the ropes, and get them into this stuff at a level that lasts a lifetime.