Hey, starting with the suspension is a great idea. It's what I did. I'm not sure if you have S13 or S14 ('95 - 98). I've got an S14, but these should apply to both chassis.
I'm not sure if the S13 has the subframe slop that the S14 does, where you push the gas or let off it and there's noticeable slop, play, jerkiness (not sure exactly how to describe it) coming from the rear. If the S13 is like the 14, the absolute best bang for your buck is a set of solid rear subframe bushings. If you're doing the work yourself, it might be a pain, but definitely do it at some point. It's a huge difference in how "together" the car feels. You could look for urethane, but I'm pretty sure I went with solid and don't have any complaints. My local shop (GS Motorsports) had them and put them in, so I'm not sure who made them. SPL, out of Dripping Springs TX makes almost everything you could want for the suspension of the 240 (and other cars). They had some listed.
A super cheap upgrade is adding Lucas power steering stop leak to the power steering system. It's more viscous than normal power steering fluid and, besides quelling a leak, smoothed out the steering feel. When I added it, some people suggested that it was too thick and I'd have pump problems. Haven't had any issues. The steering rack is hugely expensive, if you can even find one. The Lucas seems to have helped mine endure.
The next thing is a limited slip differential. While more of a drive train component, it turns the axles and the axles meet the hubs and then wheels...so, related to the suspension! THIS will probably make one of the biggest differences in how much fun the car is. The stock open differential stinks. One wheel turns. No fun.
Put in a limited slip diff and both wheels turn. Very fun. Adding a limited slip diff (LSD) will be more noticeable for tail wagging than adding more power. The cheap way, if used parts are still available (haven't looked in over a decade) was to get an Infiniti J30 (iirc) viscous LSD.
If your budget allows, or those aren't available or cost is insane from rarity, I recommend a mechanical LSD from a reputable brand.
Of the mechanical variants, I highly recommend a 1.5 way differential. This will allow full lockup under acceleration and light lock during deceleration. 1 way is only locking under acceleration and is also perfectly good for street use.
2 way fully locks under both acceleration and deceleration. 2way is great for drifting, but I don't recommend this for a street car. It makes curved off ramps and city streets with traffic lights in curves way more exciting than you'll want them to be, especially when roads are wet. Lifting off the throttle locks the diff and tapping the brakes and steering cause weight transfer and wants to send the tail around. Again, great for a track drift car, but bad for a street car (unless you like getting closely acquainted with light poles).
I've used both, unintentionally, and 1 way or 1.5 way (preferred) is the way to go. It still sends the tail around under throttle, if you want it to go, but remains pleasant in normal driving and everyday emergency situations.
Some differentials are adjustable and fiddling with how they're set up inside can adjust mode before you install. After an incident where my 1.5 way LSD was damaged due to an axle exploding as it was smashed into it (totally not a driver skill issue! /s), I'd ordered another and the part that showed up wasn't the same and wasn't adjustable. The car was already in the shop and I didn't want to wait. So, I gave 2way a shot. I don't recommend it.
I've used Cusco and been happy with the durability and not had noise issues. But most name brands are about the same cost.
If the car is stock, the front tension rods are probably bad. They use a liquid filled bushing and are just squishy. Replace with any aftermarket tension rod. I have Tanabe, I think...they've been there over 15 years, might be Tein. Current prices seem to be $100 to $300+ for the set of 2. I'd check recent reviews to see if the cheaper options hold up. It's a pretty basic piece but I've seen people break them at drift days. Replacing the stock set will take slop out of the front end.
I'd avoid drift specific high angle steering knuckles. It sounds fun and aids a track car holding long high angle drifts but I've read that they mess with the steering feel and make it twitchy for street driving.
As the other comment suggested, coilovers are great and, depending what you get, will allow a lot of adjustment to height, preload, and dampening. I'd highly recommend this over just lowering springs and new shocks /struts.
I started with a coilover set that had fairly soft spring rate from Tanabe. When the shocks needed replaced, that part wasn't being made anymore and I switched to a set of K-sport at the recommendation of the shop. They were much firmer spring rates and I was suspicious they'd be horrible for street. I was pleasantly surprised that the car rides fairly smooth except over the worst roads (where the ride will bump my head on the ceiling, but the roofline is so low, that doesn't take much).
With good coilovers, you can adjust preload, bumpstop, and dampening to get a decent ride quality at the ride height you want.
You'll also want to replace the front tie rods. Tein was a go-to for this, but SPL and others also make this part.
Depending on how low you want to go, you may NEED tie rods as well as the rear control arms (upper and lower) in order to get a decent alignment. The stock arms have very limited adjustment and lowering the car changes the geometry.
If your budget is low, I'd still recommend coilovers and then just wait on really lowering the car too much until you can get the arms to allow for good alignment. Nobody likes wheel gap, but as you said, it's a project and a process.
The order I'd suggest:
Lucas power steering stop leak
Solid rear subframe bushings
Limited slip differential
Front tension rods
Front tie rods
Front lower control arms
Rear upper n lower control arms
The coilovers, control arms, tie rods, and tension rods are all pretty straightforward, bolt on. Although, I only "held the flashlight" while people who don't break stuff did the work.
Brakes are also a fairly straightforward upgrade. R33 and R34 used to be the go-to but I don't know if this has changed now that front clips of cars might be in short supply. 300z was the typical rear brake upgrade but I only did the fronts. There's probably a few aftermarket brake kits out these days, but they were pretty pricey when I'd looked into it.
Regarding brakes and lowered suspension, when you go for new wheels, do a lot of research on what offset and wheel diameter will clear your brakes and fit under the car. You'll see a lot of what look like good deals on 17 and 18 inch wheels only to find out that the offset is for front wheel drive and will smack into the hub if you try to put them on your car.
Any 240sx or S13 /S14 wheels thread ought to have a wealth of information. I had a set of 18" that were for a 350z, and went down to 17" for a tiny bit better ride with the little taller sidewall that still looks good on a lowered car. They're Enkei and iirc I got them from Discount Tire and the offsets they said would work fit. Anything over 18" and a 240 starts looking like an off-roader unless you do things to the fenders.
I mention wheels because they're definitely a part of the suspension and great performance and style upgrade. 225 fit comfortably up front, 235 is a squeeze and the back can go to 265 width. I ran 235f/255r on the 18" and 225f/245r on the 17"
Off set widths still allowed the car to get sideways with an LSD but also a more stable grip when you're not trying to slide it. When it comes to wheels and tires, that gets really subjective but both combinations above worked well for me and looked great.
Adding power is great but modding the suspension will give you a car that's fun, more controllable, and safer to drive. I happily ran my car with just the suspension and diff mods listed above for years with only air filter and exhaust for power mods. It was perfectly capable of drifting and guys who'd done engine swaps for more power but left the suspension and diff alone were usually jealous of how my stock car could handle and slide, even if it wasn't smoking the tires.
If you're new to the 240, Drift King Keiichi Tsuchiya had a video out, years ago, with his S13, iirc. I think he had a how to drift series. Been ages so my memory is fuzzy on it. Anyway, he was big on the suspension and driver being the key components of the drift car.
Best of luck with the project and the process.