Well, Vinyl is my primary Source, so...
IMO: OP is correct: upgrade cartridge 1st.
Anyone wanting to get started with vinyl, I recommend this TT. (I had one for years). If they stick with vinyl, then move up. (what, in what order the OP asks)
https://www.audio-technica.com/en-us/at-lp120xusb
It now comes with a VM95 cartridge body, which accepts various interchangeable stylus including Microline or Shibata.
https://www.audio-technica.com/en-us/cartridges/line-series/at-vm95-series
The cartridge has a threaded body, only needs screws from top without nuts.
TT has a built-in optional Phono EQ, so it can go straight to any Line Level Input to start with any preamp with or without built-in Phono Stage.
Direct Drive, removable headshell, (useless speed control we are not DJs).
I definitely recommend buying the Microline Stylus, easily goes into the cartridge body, no re-alignment needed, no potential misalignment for a novice.
It has a lightweight platter, fuzzy platter mat, adjustable feet, nothing special there
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the VM95 cartridge body specs which relate to imaging are:
Channel Separation 23db; Channel Balance 1.5db. If you check MANY expensive cartridges, they do not have better separation or balance, and surprisingly MANY expensive cartridges are Elliptical.
Factory Aligned, Upgraded Stylus (no-realignment needed) this TT would sound very good, and I would expect the beginner would stick with Vinyl, IOW, not have a poor experience that Streaming is/was far better which happens to many novices.
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Now, time to move up:
1. Alignment Tools and Cartridge Alignment Skills MUST be acquired, by you, friend, nearby dealer before a cartridge body is changed (MM or MC)
2. The beauty of a cartridges’ sound is separate factor, subjective, but Imaging Capability involves comparable measurements.
I would first change the cartridge on this TT to get improved imaging, i.e. wider channel separation and tighter channel balance.
I always recommend this as a very good MM moving magnet cartridge to start with ($300). The cartridge/headshell combo gives you a spare headshell which will be advantageous later.
https://www.audio-technica.com/en-us/cartridges/type/moving-magnet/vm540ml-h
Channel Separation is 28db; Channel Balance is 1.0db. The combination of wider/tighter makes quite a difference when listening to a superior recording. Again, MANY expensive cartridges do not produce as wide or tight imaging.
3. Move up to a Moving Coil Cartridge and Phono Stage for both MC/MM at the same time.
NOTE: Most people would upgrade the TT first, however, the sound will be improved to a greater extent by MC Cartridge/Phono Stage with MC/MM capability. Then the MC cartridge and Phono Stage can be used later with any new TT
3a. MC Cartridge: I recommend my choice, AT33PTG/2 as a MC start.
https://www.audio-technica.com/en-us/cartridges/type/moving-coil/at33ptg-2
Channel Separation: 30db; Channel Balance: 0.5db. Combined with the advantages of MC technology.
I’ve compared it to my half-dozen cartridges, my friend’s cartridges played here, we all prefer it to the others. I’m never moving up to more expensive cartridges. I wore my stylus out, got another new AT33 at half price thru AT’s MC trade-in program.
3b. ANY PHONO STAGE: get return option, this is the most subjective move so far. Keeping it separate allows use with ANY change of equipment, and future upgrading a separate Phono Stage is independent of other equipment.
3c. OPTIONAL SETTINGS for future MC cartridges. Price definitely moves up, it might or might not be important, in my case, I had a MM Phono Stage I loved, so I added a SUT step up transformer with 4 MC optional x factors/impedances and PASS for MM use.
4. Finally: Better TT and Better Arm (OP’s Question).
4a. my recommendation: go for Direct Drive with a heavy base and heavy platter, superior isolation/leveling capability.
4b. Dust Cover. Surprisingly, many high end TT’s come with no dust cover. I cannot imagine having a superior machined Arm without a dust cover, keeps me up at night.
4c. Arm: go for a TT with an arm with removable headshell. This is difficult as so many upgrade TT’s come with Fixed Arms with pre-mounted cartridges (easy until you need to change the cartridge later).
4d.TT without arm, get/mount your own arm: this is beyond the scope of most, but a viable option if you or a friend or nearby dealer have the skills.
4d. Easy Arm Height Adjustment. Another good reason to get a TT without arm. Important IF you will have a few cartridges, their body height varies, a friend might bring a headshell with his cartridge, you might get into Mono LPs, ......
5. How you gonna keep/clean those LPs? New, Existing Used, new to you Used. Good luck with that, I stick with my manual scrub, frugal and quite successful.