New speakers for turntable ($500 range)

You have received, and seem to be taking, great advise. With a $600 budget I would consider the Wharfedale Diamond offered by Music Direct at $199 and the Denon or possibly the Yamaha A-S301 which can be had for $350. Buy some Monoprice 12ga in wall speaker cable off Ebay. The Yamaha has an internal phonostage and DAC. The phonostage will be much superior to the one in the LP60.

Once you have speaker and amp you can save for a TT upgrade. AT makes a TT, the AT-LP120 which would be a significant upgrade over the LP60. It comes with a superior cartridge also. I believe it sells for $260. Project and Music Hall also make quality budget TTs that come up discounted from time to time.

Good luck in your search.

Upgrading the amp will pay dividends with the speakers.  

The LP60 will be serviceable for now. The AT LP120 is an excellent starter TT. A steal @ $250. Significantly better than the LP60. I prefer it to the Pro-ject TTs at/ around the same price.  

But you can't do everything at once.

Don't get too worked up about cables. Good basic stuff will do you fine.  

Rich

I think mesch has given you some very solid advice. With his suggestions, you get out from under your current amp,and stay within your budget. My brother owns the yammy A-S301 and has it paired with Wharfedale Reva -1’s. It sounds very nice,for not a lot of cash. 

These go on sale every few weeks for $300. Grab them when they do. They are amazing. Check out the reviews on YouTube too. https://www.jbl.com/loudspeakers/STUDIO+530.html

Thanks all for the great advice and recommendations.  Time for me to digest it all and make a decision.  I’ll post what I end up getting.