New to vinyl - How to get the best out of old classical discs?


I’ve recently added vinyl to my classical listening lineup (mostly piano and chamber music) after realizing how many extraordinary recordings aren’t available in other formats. I wanted to start small, but now that I’m a little more comfortable with the gear and know I like the format, I’m interested in where the biggest improvements can be found. Generic suggestions are always welcome, but there are issues that seem to come up more often with of these old classical discs so if anyone has extensive experience there I would be particularly grateful. My budget is around $1000, and any gear that holds value is preferred.


My current setup:


Prep: Ultrasonic/IPA/detergent, new sleeves, quick wipe with dry swiffer duster before play

TT:  Denon DP-51F

Cart:  Ortofon Red

Phono: Pro-ject Phono Box S2


Then (usually):


MiniDSP

Scott 222C (EL84 PP amplifier)

Klipsch RF-5s (discontinued floorstanders, similar sound to Heritage)


Thanks in advance for any advice!


bluemountain
I'll agree with Chakster, the Ortofon Red is the weak link here. I think three cartridges in your price range (new) will make a significant improvement; the AT VM760SLC, the Goldring 1042 and of course the Beethoven addition of the 2M Black.
Sounds like I should start with the cart and maybe wait a bit to buy an ‘investment’ phono stage. 
Re: springs, the TT is in almost perfect shape and is obviously a masterfully engineered machine. I personally wouldn’t try change anything about how it’s put together.
Add the  moving iron Soundsmith  Otello for $399 with the other suggestions.

It can perform on the level of carts twice the price. Consider upping the phono stage for your long game. Better phono stage will be ready when you go to the next level table. 

Non stop madness if you choose to play with rekkids. You're NEVER done!
Re: springs, the TT is in almost perfect shape and is obviously a masterfully engineered machine.
Adding springs under the table won't impact it. Try them and return them if the don't work for you. I bet you'll be real surprised.
The bottom line is the source. Some records are just not well recorded/mastered. If you had the right disc, recording, and pressing you would not need these suggestions.