What is the order of importance of the parts of an analogue rig?

Hi,
all have to be of good qality but if order is needed, turntable, tonearm, quality of lp, cartridge, phono stage, cables.

Where can you get the most improvement from a change in one category?

From whichever one is the worst. Sorry, but its true. And I know, from having done. I’ve actually built and upgraded and modified each and every one of the things you mention. Except mats. Mats are for .... well never mind some guy somewhere swears by mats so let him be. A mat though is really just a convenient way of changing something without going to the expense of changing the platter, which if a mat helps is what really needed improvement.
But really every single one of these is equally important. Some guy somewhere changed something one time and it made a bigger improvement than another thing he changed another time and so he leaps to the conclusion Oh its that thing! Wrong. Its all of them.

What questions like this usually mean is What should I do next? What’s the biggest bang for my audio buck? And there is no answer to that one either.

(You will get lots of answers. This site exists to list things guys should buy. Watch and see what I mean.)

So with turntables I look at things a bit differently. The components all are equally important, but they are not all equal. The cartridge, unless you get something like a SoundSmith Strain Guage most cartridges are pretty much disposable. Tone arms and turntables on the other hand are pretty much forever. My turntable for example is 17 years, arm about 12, cartridge less than 2.

So I throw out which is more important. Instead what I do is determine which one might be the weakest and make it the strongest. Started with the arm, but there is no particular right or wrong order. If you find an awesome table at a good price just get it, mount your whatever arm, enjoy. Then if you find your dream arm and have the dough put that on there. You will indeed hear and appreciate each one. Ditto cartridge.

Seriously, this is for real. I have compared different platters on the same table, heard the difference. Changed bearings and even individual parts within the bearing. Changed motors, motor controllers, motor controller power supplies, on and on and freaking on. Belts? Changed em. Clamps? What did I just say?? Lol!
https://systems.audiogon.com/systems/8367
So don’t sweat it. Just do it. Whatever it is, I can just about guarantee you will be telling yourself wow that was the best thing I could have done! Until the next one....

I'll say the most important is a cart, that you like.

Everything else is a "need to have". To what degree?

I like real heavy EVERYTHING.. Plinths, EXTRA heavy!!

Being clean... Some folks are, some folks aren't...
LPs, take care of them..Handle with care!!!

Vibration control is a BIG thing, for LPs, ay?
It is # 1 right? or close...

Regards

1. Cartridge, but you need to have it well matched to a proper tonearm that won’t cripple its performance. Or else you goofed. But always start at the transducer since they impart the most coloration and your sonic preferences here will be very personal.
2. Phono stage, which needs to be electrically and sonically matched for your chosen cartridge. Includes the SUT if you go that route for MC step-up. Some people can tolerate more or less noise here which will factor into your decision. Choose poorly here, and you’ve also goofed.
3. Turntable needs to be good enough to not destroy the performance capabilities of the above items. This includes necessary isolation from your floor, speakers, other energy sources, etc. Choose poorly here, and you’ve also goofed.

Mainly remember your’re trying to put together a complimentary system of well-matched components. E.g. you can’t go "all in" on a Koetsu cartridge and stick it on a Project Debut. That’s why when it’s time to upgrade, you may find yourself buying all new things, which is why it’s such a money pit :)

Anyways, the way most people buy into their first vinyl system won’t be so orderly. Many will go for the bundle packs, like Project or Rega, with varying degrees of success - though at least the components will be reasonably complimentary, and you just need to choose phono. But if you want to do this "right", you’ll have to spend more than you first think. I started 13 years ago when I fell in love with a consignment SOTA Star III in fiery Koa wood, and just had to have it. It came with a Fidelity Research arm (great arm) and fortunately I didn’t cheap out too bad on the cartridge - got a new Benz Micro Glider (perhaps a slight mismatch to the arm, but not too bad) and a Dynavector P75 phono stage (I could've done a bit better here, but not without spending more). It cost $3.4K all-in, and the setup sounded so good I became a vinyl junkie for life. When I had started looking, I was hoping to keep it down to low-$1K max. Whoops - but I’m so glad I invested much more.