SEEKING ADVICE: Which component would you upgrade first?

A few options from someone who has been down your road recently. Not sure I would say any of these is the "right" way to go, but any of them will make an audible difference.

One, I don't have any experience with your receiver, but if you're thinking about upgrading your amp the NAD C316BEE is a great option. It brings that classic NAD class AB sound, and has a really good MM phono stage.

Two, I'll echo the previous post about upgrading your phono cart. A big improvement can be had by upgrading/updating to one of the Sumiko oyster carts, the Nagaoka MP series, or the AT VM that was previously named.

And third, if vinyl is your primary media, think about a record cleaner. I use the Record Doctor, which made a big difference in audible quality of my vinyl for not a huge investment. And if you play a lot of used vinyl, an inexpensive ultrasonic cleaner makes a remarkable difference. 

Good luck and enjoy the music!

Corelli +1, If you know an audiophile near your place try to listen to his system , to see if you like it. Then if you like it. Hopefully he can mentor you to put a low cost system if you don’t have the budget.

cheap yet HIGHLY effective: Mad Scientist Graphene Contact Enhancer

First clean all contacts (including power cords) with 99% isopropel alcohol then apply the contact enhancer www.madscientist-audio.com/contact-enhancer

hth

If you want more modern electronics, look at Rotel. Modestly priced, good quality, and nice features in their integrated amps. 

@lukasread I’ll suggest a different route to consider, if not now then at some point, given you said you have a background in music engineering and an interest in vinyl. This route would have you listening to your music in other ways, rather than swapping kit-for-kit in hopes of sonic revelations for a few hundred bucks.

There’s a USD $250-300 device (down from $500 nearly a decade ago) that functions as a phono preamp, a DAC, and an ADC (Analogue-to-Digital Converter). RCA cables from your TT to the device, and the device via USB cable to your computer; and you can digitize all your records to enjoy on the go via earbuds, in the car, on your phone, etc.
KORG DS-DAC-10R

The DAC feature of the KORG will also allow you to connect a USB streamer that can be had for ~$100, or easily DIY’ed for slightly less. There are multiple free software options for this that will also play your personal digital/digitized music library, too (though as a music engineer I suspect you may already have these features via computer / additional unlisted kit).

If you want a way to clean your records and haven’t one already, there are cheap, solid solutions from AliExpress and the like:

Record Spindle Spinner (I would strongly recommend a stronger jet of water than pictured in the article!).

As some folks have mentioned, beware making lateral moves in the kit you already have - yours is solid stuff. For your budget, look for new inroads to using your music if you really are a musician, engineer, and listener more than a new-kit-chaser, being my bottom-line.