How to reproduce sound of piano

I agree with Shadorne on the ATC Actives. Very convincing in their reproduction, and because they are active, you can buy a minimal amount of gear to support them. I prefer Bryston solid state pre-amps with them and you could keep or enhance your source components.

If you wish to migrate to a warmer, tube sound: Two other tube systems that sound great are top-of-line Opera speakers with Unico Research tube components, or the new Harbeth 40.1 speakers with a good Japanese tube company (audition Leben, Shindo or Almarro.)

The sound of the Beveridge is delicate and well defined but it lacks the dynamics one would wish for, feeling the need to get it more or less right.
I've long lived with a Quad 63 and Gradient sub combination, driven first by Jadis gear and later by two sets of Atma-sphere 60 watt monos and an Atma
MP-1 Mk. III phono pre. Timbre, presence and subtle cues were there, dynamics were fair, but not as right as I would have wished for ( I am initimatley familiar with the sound of a grand ) and the lowest octave, when "thunder" was called for, lacked both power and presence in spite of the Gradients, which performed fair otherwise . All the same, the middle octaves of the grand were there, the very highs almost and the lowest lacked punch. Liszt made me not quite happy, Schubert or Mozart mostly completely. I mention this, because if you buy used, you could perhaps make it, including a good TT plus arm and cart. I am still waiting for good piano rendition on digital. If someone could point me to good recordings of classical piano, I would be grateful.

To get it really right is not cheap. I am at a point where I am truly happy, but at what cost!
To wit:

TT.Goldmund Reference I, highly modified with space age plinth and pivot.
Arm: 1.Highly modified Goldmund with adjustable VTA from listening position
2.highly modified Souther with Air Tight PC-1 (best for dynamics)
other cart: Suchy's Goldfinger on the Goldmund arm
Phono pre: Wavac and Boulder 2008
Pre: Wavac
Amp: VTL Siegfried monos Pentode mode (700watts)
Speakers:
High: aCapella plasmas driven by Zanden
Main: Sound Labs U1-PX, wide dispersion
Bass: 2 Soutien self powered bass systems with room control
Sub bass: Rel Studio III
Wires: Stealth and custom made.

I bought much of the above used or at discount, but all the same it came to well over 5x your budget. Perhaps you can get the same result for less money, though frankly I doubt it, if you want to get satisfactory results from top to bottom octave, from ppp to fff, have all the resonances and decays right, knowing exactly how a Steinway, a Bösedorfer, or a Bechstein with what music can sound and be enabled to sometimes almost forget that your're only listening to facsimile of the thing.

Wow thanks for all the responses. It appears that the task is much more daunting than I imagined, but that makes this hobby the more exciting. FYI, my current system is in my condo but I'm planning to build a dedicated listening area in my weekend house so the space is no longer a factor. So from so far, looks like right speakers with good subwoofers and a top notch turntable is the way to go.

It probably goes without saying, but since no one has touched on it yet; not only is the Piano one of the most difficult to reproduce, it also seems to be one of the most difficult to record judging by the wide range of results I've heard on my own systems. The recording itself is going to have a profound effect of how real the instrument sounds. Seems to me that in general studio recordings are more successful than live stage recordings in bringing the instrument into my room. BTW I'm not using the classic formula of profound muscle from my amps to do this. I use 9 watt SET amps and high efficiency speakers in a moderately sized room. It's certainly not the last word in 'live' sound, but it does a pretty good job to my ears. I've also had powerful amps (Bel Canto Ref 1000's) in the same room and they did a very nice job as well, but I prefer the sound of the SET amps overall. I do use a sub. I know you said you listen to mostly classical (as far as piano music, so do I), but the most realistic portrayal I've heard of a piano in my space comes from a gifted alt-pop artist, Tori Amos, who was a child-prodigy on piano and remains an amazing keyboard artist. She is a Bosendorfer sponsored artist. Pick up a copy of her out-of-print EP of "God" and listen to the two piano solos on that EP. That's the closest I've heard to the piano in my listening room. It occurs to me as a very closely miked recording and has the most immediacy of any I've heard of a piano. Most of the classical recordings I enjoy seem more distant in comparison. I wonder if anyone could comment further on this - why aren't there more recordings like this in the classical realms?

I heard a pair of Audio Note speakers at a dealers'. I wasn't there to audition and it was very brief but I clearly remember how life-like the piano sounded.