Turntable versus tonearm versus cartridge: which is MOST important?

$500 can get you a nice cartridge, it does not have to be something unworthy of a nice table.  You can then upgrade the cartridge in time.  It is far harder to upgrade the table or arm, unless both are purchased at the same time. 

When looking for that nice table, look for models that put an emphasis on the arm.  It is getting harder to find decent arms these days (a major OEM maker for a number of brands--Jelco--went out of business).  The arm will become quite important when you do upgrade the cartridge; many of the better cartridges are lower in compliance (stiffer suspension) which means they have a tendency to feed back more energy into the arm which has to be stiff enough to handle this energy and the bearings musst also be stiff enough not to rattle while being low enough in friction for their movement to be unrestricted.  

Perhaps it is not as extreme as this hypothetical, but, I know someone who bought a $5,000 cartridge to temporarily use on the cheapest Pro-ject table.  I helped another friend with the installation, which included milling a heavier counterweight to use than the one supplied by Pro-ject.  The installation involved using a Feickert protractor, and a very good digital scale (I own equipment for setting azimuth electronically, but, we thought this is overkill for this installation). The combination sounded good, except for some noticeable inner groove distortion.  The distortion was probably from the arm being barely able to handle the vibrational energy from the cartridge.  When the cartridge was installed in its permanent home, an EMT turntable/arm, there were no issues with inner groove distortion.

Like everything else in audio, there are extremely expensive examples of tables ($500k +) and tonearms ($70 k +) and cartridges ($20 k).  But, I think you can get quite close to state of the art sound at MUCH lower prices with a good table/arm/cartridge selection than you can with comparably priced speakers and amps.  For the $10 k budget, you do not have to make big compromises in sound. 

A local dealer who has assembled system well north of $1 million, and has many customers with systems costing more than $100,000, will recommend as worthy tables reconditioned Thorens 124 and Garrard 301 or 401 tables with Ortofon tonearms.  Sometimes the cartridge recommendation is pricey (Koetsu Rosewood Platinum), but still, the combination is much less than recommended speakers or amplification.