New speakers for turntable ($500 range)

An almost 'impossible to answer' question given the way you worded it.

If I understand it correctly, you wish to spend $500 on speakers for an amp that costs $90 and the most positive of reviews for the amp on Amazon are about how fast customer service gets back to you when the amp craps out, which it seems to do often enough.

So, don't go overboard price wise with anything with this amp as your system base.   

First things first, it sounds like you may need a phono preamp for your TT with this amplifier, unless your TT has a built in phono preamp.  

The Keiid is a 40 wpc amplifier.  No idea over what frequency the wpc were measured, so no idea if you have 40 wpc continuously available to you.  Also, no idea as to what type of music you listen to or how loud.  I have never heard a Keiid product, I am shooting in the dark. 

I would limit the speaker purchase to $200 and go for something like these:

https://www.musicdirect.com/speakers/wharfedale-diamond-220-bookshelf-speakers

https://www.soundstagehifi.com/index.php/equipment-reviews/911-wharfedale-diamond-220-loudspeakers

https://www.whathifi.com/us/wharfedale/diamond-220/review

Wharfedale makes a good product.  You would most likely keep these if you upgraded the amp.

Keep in mind that they are only going to play so loud, but they will sound nice.  You are not going to get all the subtleties that are available to you, but you will enjoy what you do get.

I have setup some inexpensive systems that sound real nice.  These days, I suggest powered speakers (Kanto) and TTs with built in preamps AT or Music Hall).  For small apartments they do very well.  In bigger spaces not so well.  

Good luck.


Rich 

On the plus side, you get it about sound quality and appreciation of what sounds good and what doesn't.

It sounds like you have a limited budget for the time being ... so what to do first?

So, what to do if there is only $600 to spend?

I would stick with the speakers I originally suggested.  They have been well reviewed.  The Wharfedale house sound is pretty pleasing.  They will look good in your dining room.  

https://www.musicdirect.com/speakers/wharfedale-diamond-220-bookshelf-speakers

The Keiid amp will limit you.  

Something like this integrated from Denon could work (Denon PMA 600).  It also has a built in phono preamp, which will most likely be better than the one that comes with your TT.  It has Bluetooth, as well.   

https://www.bhphotovideo.com/c/product/1521880-REG/denon_pma_600ne_integrated_amplifier.html

At a $100 more, this Denon (Denon DRA 800) has been well reviewed.
https://www.crutchfield.com/p_033DRA800/Denon-DRA-800H.html

At this point, prices start to creep up.  I also did not suggest buying used, as I prefer new.  

If this all works, the TT would be the next candidate for upgrading, but that can wait.


Rich 

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