You can get there from here, but beware of "smooth" sounding gear. Much of it sacrifices detail and imaging to eliminate harshness, particularly tube gear, but not all.
If you want to retain all of the detail/imaging and dynamics while eliminating the harshness that often accompanies older CD recordings, you will need five things:
1) reduce jitter from the digital source as much as possible
2) eliminate as much digital filtering effect as possible
3) eliminate the active preamp connected to the DAC and use either a new technology volume control in the DAC or use a transformer linestage such as Music First
4) eliminate as many ground-loops in the system as possible, both analog and digital
5) regulate the AC power to your digital components
Your budget is possible, but it will probably run over to get all 5 of the above tackled. However, if you accomplish even 1-4, your old CDs will be joy to listen to.
One set of components that will do 1-4 is the Empirical Audio Synchro-Mesh reclocker driving the Overdrive SE DAC driving a set of Final Drive transformer buffer/selectors. Total in the $9-10K range.
The Overdrive allows selection of digital filter roll-off, so the 192 filter can be used when playing 44.1 tracks (CD). It also has a volume control technology that is superior to even the best preamps because it actually connects the DAC line-out directly to the amp. This is not a digital volume control using DSP software. This is a real volume control. The signal distortion actually decreases as the volume is lowered, unlike any other volume technology.
http://www.empiricalaudio.com/products/overdrive-dac
The Synchro-Mesh is a reclocker you can use with any CD transport, expensive or cheap. Even Sonos or AppleTV. Does not matter. It delivers an ultra-low jitter signal to the Overdrive SE DAC no matter what feeds it. The jitter in this signal is actually more important than the DAC, so this is a critical element.
http://www.empiricalaudio.com/products/synchro-mesh
The Final Drive inserts after the Overdrive SE DAC and will output either RCA or balanced XLR and allow you to insert and select analog signals from a vinyl or HT preamp and still drive your amps directly from the DAC, avoiding any preamp. It breaks the ground-loop between DAC and amps.
http://www.empiricalaudio.com/products/final-drive
This configuration has earned awards (best of show) at RMAF and Newport for several years. Several customers have even sold their vinyl rigs after getting this combo.
Tackling #5 is harder. The only device I have found (and it works fantastically) is the Plasmatron3 from VHAudio.com. Around $3K. I don't do any digital without it anymore. Makes that big a difference. This can always be added later when the budget allows.
If you want to put some effort into computer audio, this will be even better than the Synchro-Mesh driven from a digital source or CD transport. More expense as well.
Steve N.
Empirical Audio