New speakers for turntable ($500 range)

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.