Small room amp and speakers recommendation

  Have to admit, Mr. crusty's thoughts on the DIY fest with the Akitika gear makes me want to get out my soldering iron!

I would look into the Salk Song3 speakers with a Prima Luna integrated. I have this setup in a smallish space and it's great.

You haven't mentioned if you can get the speakers away from the walls or not. There's some great recommendations here, but some I'm not so sure about since you're not going to have much in the way of fudge factor. JBL and KEF are both good for near field type listening.
There's some interesting recommendations, I'll come at it from the cheaper end, as I always start low and work my way up. There's been some great integrated amps mentioned, so many good ones out there you really can't go wrong. Don't overlook companies like Schiit, that make budget and space friendly gear. Currently, you can buy the Polk LSiM 703 bookshelves for less than half off - about $600/pair. All the good Christmas sales are over - like LS50's for $900. At this point I'd never pay $1300 again.

Further improving your available space, I'd consider Elac Navis ARB-51's and kill 2 birds with 1 stone. That gives you a lot for your turntable/cart/cable, and a bit left over for streaming if you want to go there. Plus, the concentric design works incredibly well for near/close field listening. I prefer them a lot to something like the KEF LS50W's. For even less money, the Edifier and Fluance have been getting rave reviews. I've heard lots of positives from those liking jazz and classical.

You could go with Fritz Carrera 7 BE. In home audition is available. It’s not too hard to drive, so plenty of amp options. At Capital Audiofest 2018 they were paired with an 8 watt Modwright headphone amp
https://m.youtube.com/watch?v=zbroWg_cwCQ
That amp is above your budget, but I just mentioned it as an example of amp choice flexibility. Plenty of nice reviews out there for this speaker.

Thanks so far for all the advice, I have been reading a lot over the past 2 days. I was kinda hoping a few speakers and amps would keep getting recommended so a could focus my search, but I can see just how subjective audio equipment can be.
I am going to research more into the sizing of the room to try and tweak the acoustics as much as possible. 
I know I will be running vinyl on this system, as I have a brand new project carbon espirit turntable, other than that, I assume eventually I will stream of get a good cd player.
Overall, I want to try and keep the system simple, 2 channel amp, speakers, good interconnects and the turntable...although I am probably missing something.