From the looking around I've done, it seems like a reasonable "starter" headphone setup is to get the Massdrop/Sennheiser HD-6XX and a Schiit Magni/Modi stack, but that is ~$400 and hard to stomach for a hobby I'm just getting in to.
Here's the thing though.
For any hobby or sub-part of any hobby where there needs to be some use more than collecting and displaying there will always be these ancilliary costs. For example you don't necessarily have to collect paintings that you need to buy expensive insurance for while keeping them in a room with laser and radar motion sensors. You can for example collect Gundam model kits, and you only need to use say your nailcutter (unless you're too grossed out by that) to clean the snap off plastic points on those 1/144 kits and then an Ikea shelf with some blu tac to display them. Once you get into the point where you're making dioramas, usually recreating scenes from the anime or manga, that's where it gets expensive. For a turn key solution there's a 1/60 size kit of the Impulse Gundam that comes with a spare Sword Silhouette (one of three equipment packs) so you won't have to buy two 1/60 size Impulse Gundams in two different colors just to get the sword equipment pack, put it up flying near the multi role combat pack equipped Impulse Gundam, and then have one of the anti-ship swords in the hands of the otherwise mutli-role equipped Impulse; but then to complete that diorama you're gonna have to get a 1/60 size Freedom Gundam...but then to make it look accurate by that frame on the anime, you're gonna have to start breaking wings off the Freedom, maybe even get a soldering gun to tear a hole through it and stick the Impulse's anti-ship sword through. And at this size any tiny imperfections like improperly lopped off and unsanded connections from the plastic holding the parts in the box need to be smoothed out, so at minimum you're going to invest time if you're not grossed out using the same toenail cutter you use on your actual toes. Want realistic battle wear? You're gonna have to buy a small paint gun and some grey and black paint. Even if you recycle foam to make it look like the sea and beach where this scene was you're still investing on the kits and the equipment.
And then here's another one of my hobbies: cooking and its sub-hobbies aka each equipment type and the food. So for example you can't just get a good quality steak and sear it on a cheap teflon skillet - the only cheap skillet you should be doing that on is cast iron, where the cheap cost is offset by the weight and the maintenance (which for some people is too much, for some reason). You can just grill it, but then that assumes you live in the suburbs and have a backyard instead of living in some small flat downtown that you only managed to afford because you're not paying for a car and there's rent control. And then even if you're on a cheap skillet, normal oil is fine for keeping it from non stick and rusting, but if you live in a relatively humid environment, you either have to make sure you cook often or you have to use an oil that doesn't grow rancid as quickly as cooking oil while still being an edible oil (because, ya know, food), like rapeseed oil, which might not be easy to find in some places.
And then there's knives. A friend asked where I got one of the knives I was using when we were on vacation, how much it was, etc, and he was surprised I got it for $26 off an Amazon retailer and wanted to get one even at its regular price of $56. But then I told him it doesn't end there. It might be cheaper than my other knives that I use at home or when I don't have to board a plane, which is a kit with veg chopper, a paring knife, and a slicer for raw fish that I learned to compensate for when I have to cut through roast meat (because the dedicated slicer is going to do just that one thing and I'm not blowing money on that), but he's still going to have to buy comparable whetstones that I use on those knives. Why? If he uses the kind of stones from a hardware store that isn't for woodworking hobbyists or from a department store he'll end up doing one of two things: a rough edge even on a two sided stone or not knowing whether using only the smoothER (ie still not fine enough) side on the cheap stone is just taking too long to get it sharp or there's something wrong with how he's doing it, unlike my being able to tell him that if he uses the same 1000grit Suehiro that I have he should take only about six passes without forcing the steel against the stone with enough force to dish it in the center. If he's dishing that stone he's pressing on it too hard; if he's at twelve passes he's at the wrong angle. If he's using a cheap stone, I can't tell if getting to 24 passes and not getting a result is normal for that stone, and even
if that was normal, why would anyone want a relatively rough edge out of sooooo much time and effort. I told him he's better off buying a cheap knife he might have to sharpen more often or stick to a knife he can get from a local store and have it sharpened every month or so...which within a year ends up costing a lot more than just getting the stones or hell, his own motorized sharpening tool (the only reason I don't even have these over two stones plus a strop is because they take up waaaaaaaaaay too much space; if I had a suburban home I'd have this replace one of those stones, which happens to be the one that wears out faster). Then on top of all that my not ridiculously expensive knives are all bare steel with no chromium, so hohoho, he's asking why I kept wiping them dry after chopping half an onion or slicing up the fillets...all because they'll rust while a comparable hardness stainless steel knife would be more expensive than the ones I use.
Back to audio though...the problem here is getting the Sennheisers. If you opt for those and don't get an amp, it's hard to flesh out whether you just don't like what they sound like or they just sound anemic without an amp, let alone something like a K702 which will have problems other than an anemic low end. The recommendations you get at that point will be to get an amp so it would be clearer, ie if you like them then it's the amp, if you don't then it's the headphone; or to get a different headphone that won't need as much voltage (or current for other headphones), which personally you can just do in the first place.
Note: nothing is foolproof so even if I suggest getting an AKG K553 Mk2 instead so it won't have as much of a need for an amp, that doesn't mean
1. You'd like the sound
2. That they'd be close enough to the sound of an HD600 for example, especially if they fit a little loose on you (the closed back and leather pads might compensate for that though...
might)
3. That they'd sound completely fine on whatever you plug them into (assuming they don't fit too loose that the bottom end of the range is just not there anymore)
My budget is $100, with a stretch of I'd go up to ~$200 if it seemed like spending only $100 would hardly go anywhere. Do you have tips on a good way to use that money (likely in combo with stuff I already own?) to have a better experience listening to music?
Maybe the K550 Mk2. Or just get an IEM with BA drivers so they're not as susceptible to high output impedance issues, just not too high sensitivity with too low impedance so they won't hiss either, but also make sure the eartips you use have a tight fit. Something like a Westone UM20 Pro, or something similar but cheaper.