New to vinyl, where to start?

Hello all,

I was recently gifted an Audio-Technica turntable, and I would like to get started with a vinyl system. Previously, my “audiophile” set up consisted of an iFi DAC that I found on Audiogon and a Sennheiser HD569. Maybe not quite audiophile sound, but very effective for my student-sized budget. 

My question is this: given a budget of $500, how can I get the best sound out of this new turntable? My initial thought was to find a used HD 650 for ~$300 and a good tube amp for ~$200. Is this a good place to start or should I consider other elements such as updating the cartridge? I’m new to the world of analog sound so I would appreciate any advice you all might have.