New cd player-- or should I get SACD?

My family has large CD collection and old Sony cd changer. We agree it sounds digital.
Looking for new digital medium player for my room. There are 1000s of recommendations on Audiogon, I can't listen/try all.

Analog sounding (non-digital) for classical music (non-opera).
Should I get SACD player? Budget is $2000 for digital player.

My question is, is it worth it to search for a cd player, or should I skip to SACD? Some said SACD players don't do redbook well. Should I get separate CD and SACD players, or are there units which do both well? I am interested in playing both.

Universal player...see Marantz or Denon models in your price range. Redbook will be quite good (much, much better than you are used to) with the added SACD and DVD-A capability.
Since you do like classical music I would say go for a modified Sony, Marantz, or Denon, SACD player from one of the reputable Modders (Underwood, Modwright, RAM, etc.) I say this because if you are a classical lover the RCA Living Stereo and Mercury Presence SACD collections (50-60 discs and counting) are worth the investment in SACD alone. And since you are going with a modded player the redbook CD will sound excellent too. There is lots of great classical music in SACD and there seems to be more coming out.
I think you want a "Universal" player. With a SACD player the PCM D/A may be a second-rate afterthought. If you have a player with DVDA capability this means that you have an up-to-date 24 bit D/A that can be run up to 196KHz, and that can't hurt CDs.
I bought a used Musical Fidelity A5 player here on A'gon for $2000. Couldn't be happier, it was a huge leap forward in musicallity from my Marantz CD67SE with MF X10D tube buffer. Lot's more air, soundstage and timbre rightness. I found the JPS digital AC power cord to be a terrific match for the player, they can be had for about $250.

There's another A5 for sale here for $1900.
I haven't considered SACD players, too little software that interest me there.
I agree with Pardales that you should get a modded player. I suggest a modwright sony ns999es with all the mods (see his website - which is slightly outdated.) which you should be able to pick up at your price point on audiogon. I would do your research on modded players. I recently purchased a Denon 3910 and had Dan Wright modify it. It soulds great on both SACD and redbook and can be used in a HT system to boot. Also, Dan is very professional. If you have any questions, I would not hesitate to call him. Also see Robert Levi's glowing reviews in Positive Feedback.
If you are willing to spend an additional $500 you could probably buy a lighly used sony ns 999es on audiogon and send it directly to him. The turnaround time on my unit was less than 10 days.
Good Luck!
Don't throw your money away on a new CD player, or worse a SACD player ( I know that this is sacrilidge but, let's be honest, SACD is a dead horse no matter how much better you think it is audiowise).
Put your music on a computer, either a powerbook or a mini MAC, and get a DAC from here:
If you don't use compression when putting your music onto your computer it will be just as good as the original CD.
The CD players will be seen in the near future as we see cassette decks now. Something that used to be our favorites but, are now thrown aside. And on the digital side of audio, the future is already here.
As to your willingness to buy both a CD player and a SACD player so that you can have a "analog" sound for your classical music. Someone has been blowing smoke up your skirt. SACD is not analog sounding, no matter how many times Sony tried to sell the idea. Sometimes, on some SACD's, it is sometimes better sounding than CD's. There is one thing that isn't sometime though, you get to always pay more for the SACD for the privalige of hoping that it might sound marginally better.
If you are willing to buy two players and new software for your Classical music. And, you want analog sound. Then make the second purchase exactly that, analog. I won't get into the ridiculous debate of which formate sounds better because they are both good. Some CDs are better than some records, and vice versa. But, and this is a huge but, vinyl is more fun than digital. Now why would you want two digital formats when you can have a second format that is a whole different experience in both equipment, software, listening, and did I mention that vinyl is twice as much fun as digital. You get stuff to tinker with, you get software that is actually fun to hold, you get real album covers and album art, and did I mention that vinyl is ten times more fun than digital? Just so some digital only nerd doesn't jump on the thread, yes, vinyl sometimes has surface noise, sometimes it needs to be cleaned but, did I mention that vinyl collecting and playing is 100 times more fun than digital?
So, no CD player. Buy into the future, get a DAC that can be used with your computer and start loading up all that music you own. It will give your kids something to keep busy doing while they talk on the phone. Then, for when you want to have fun, as well as listen to music, get a record player.
If u listened to classical most of the time, then go for a SACD player.

If not a redbook player will do.
I say stick with the Red book CD's and put money into a really good Redbook machine. I have both, a Sony DVP NS999ES which does SACD and a Naim CD5i for Redbooks. There are some really good sounding SACD's out there, but on the whole I actually get better sound from the Naim with RBCD's than I do with my SACD's. It all boils down to the quality of your source player.
Get an SACD/CD player or a universal player. Get it modded and you will be very happy. You could look for a used one already modded for a great deal here on AGon also.
If you are a classical buff, there is lots of software for you out there.
I have about 20 carefully selected SACD's and they are a step above redbook in most cases.
I still listen mostly to redbook, but it is nice to have the SACD option.