New to vinyl, where to start?

You can possibly better the sound by using an external phono Amp, like the schiit mani or even an ifi phono Amp. Your headphones are fine and have pretty good reviews. To improve those, maybe a decent headphone Amp. As others have mentioned, that cartridge, which I assume is AT, can be upgraded just by swapping in another from the same AT line. You do not mention an Amp and speakers...but you do say you have only 500 bucks. It would be hard but not impossible to assemble something of quality together at that budget. You could look at the cheaper Cambridge Audio stuff like the AXA25 model integrated , $299/Amazon, paired with maybe some wharfedale’s. The diamond 220 speakers are still available for $249/Amazon brand new, great speaker for the money. Plug a schiit mani (about $100) into one of the line inputs of the above AXA25 integrated and you are all set. If you can scrape up another $150 above your 500 buck budget ($650) you can get the 3 above components and have a nice little set up. Records of decent quality vg+ can easily be found at the thrift stores. I scored many a good record from them. I’d rather have many records to play on cheaper set up, than too few on an expensive rig. Grab a spin Clean record washing kit, a carbon fiber brush, stylus cleaning brush, and some quality replacement record sleeves and you are good to go on that front. Then enjoy the music! Vinyl can be very rewarding...

You will NEED a phono preamp

The best advice you can be given is to not put the cart before the horse. As you upgrade your equipment, the amount you spend should be at least equaled by the amount you spend on LP's. You don't want to become an audiophile with 100 records and a $2,000 record player, and it takes only 100 $20 records for that to become reality!

Do you know how many hours the cartridge has on it?