Upgrading my beginner setup and not sure how to allocate funds. New cart, table or amp?

Start with a cartridge, this is the most important anyway, your current cart is entry level, you will have to change it on any turntable, also you can buy it cheaper than amp or turntable and on your TT it will be immediate improvement in your system. A killer cartridge for your AT turntable or on SL1200GR is Stanton 881s mk2 or Pickering XSV3000 (both with stereohedron stylus) that can be found for under $400 easily. Higher models like Stanton CS-100 or Pickering 7500 are in my top-10 MM. Audio-Technica high-end MM in AT-ML series are great too, something like AT-ML150 or higher.

I haven't heard your integrated or speakers, but unless they are horrible, I would upgrade your turntable.  Most turntables come with a decent cartridge, so you can kill two birds with one stone. 

Right at your price point, I'd take a hard look at the Marantz TT 15S1.  It's very similar to the Clearaudio Emotion, which I own.  It comes with the Clearaudio Virtuoso cartridge (which I also own, the wood version) which is a really nice MM cart. 

Another option would be a used Music Hall MMF 5.X or 7.X turntable and a Hana SL cartridge if your integrated will handle MC carts.  If you're happy with the speakers and integrated, that would be a big upgrade. 

Looking at it another way, speakers will change the sound of your system more than any other component.  Making an assumption based on your current speakers that you don't have a large space and you like monitors, you might consider a used pair of Harbeth P3ESR speakers.  You should be able to find them used at your price point.

I am looking to spend roughly $1500, though could maybe spend more if I stagger my purchases (i.e. buy a new cart now, buy an amp later, etc.).

Staggered upgrades- great idea! That's how I did it. 

If you do decide to go with the cartridge first then do yourself a favor go check out Soundsmith, listen to Lederman and his fixed coil design. Both because they are super value in a cartridge, and also because they are super long term value in that they can be retipped for a fraction of what everyone else wants. All other cartridges are pretty much consumable items. Only Soundsmith with his retip/rebuild program is one you can consider as long term a component as the turntable and arm.

New TT for sure. That's the absolute weak spot IMO. Don't know how old your integrated is but I'm guessing it may be 20 yrs old or more. If so, the have it recapped (new capacitors). I did this to a 22 yr old Conrad Johnson PFR preamp and it was the best spent $200 I can remember. Its only 20 wpc but if your speakers go loud enough for you than that's fine. 20 wpc is plenty for 93 db speakers but yours are close to the cutoff point. An MMF 7 Music hall with the project speed box would be a good choice. Also, Id guess you may lack bass. If so, you may have enough budget left over for a good sub woofer. Bear in mind, I have made a lot of presumptions. Only you know if the amp and speakers work together relatively well

Thanks, everyone, for the great feedback.

The assessments of my current setup are mostly spot-on. I have a small listening space, and the Triangles + NAD together are a nice combo for for the folky (acoustic guitar & vocals) stuff that I listen to, but are definitely lacking when it comes to a more dynamic "rock" sound. That is, there’s very little low end and it often feels like the the speakers aren’t being pushed (if that makes any sense). Oddly, It really hadn’t occurred to me to replace the speakers (maybe one day a pair of Dynaudio bookshelf speakers?)

There are quite a few recommendations for cartridges in this thread:

• Ortofon 2M Black
  
• Nagaoka MP110
  
• Stanton 881s mk2
  
• Pickering XSV3000
  
• Soundsmith
  
• Clearaudio Virtuoso
  
• Hana SL cartridge

Is there anything that I should be mindful of if I decide to buy a cartridge for my AT-PL120 with the intention of then using it with the next table (which is feeling more and more like an inevitable purchase)? Or, is the issue mainly the cartridge’s compatibility with the amp/phono stage? Further, and I know that this is a broad question, but how the hell do I narrow it down?!

Ultimately, I think that for the turntable I’m deciding between:

• Technics 1200GR (upper end of budget and doesn't appear to be sold w/ cartridge)
  
• Music Hall MMF 5.3 (comes with Ortofon 2M cartridge)

Williewomnka’s recommendation above that I check out the Simaudio Moon 110LP phono stage has me thinking that perhaps a modestly priced phono stage would be a very worthwhile investment as well. Perhaps pairing that with the Music Hall 5.3 (~$1500) and a ~$500 cartridge would drastically improve my setup.

I do feel a bit in over my head but your comments have been encouraging and informative. I am very appreciative of your time and input.

Tyler