The car buying market is a huge mess right now. My mother-in-law wants to buy a new car and my husband and I are doing the footwork (checking out cars, getting quotes, etc.) I'm not sure what car you're planning to buy, but many are either out of stock or have been hit with the impuestazo as it's being affectionately called. There's problems with parts and imports. For example, a dealer told us there are no more Renault Dusters until next year... and maybe not even then. Heard the same about a couple other models at dealers. Some say they are waiting out until taxes/rates go up again (which supposedly soon), but who knows.
As for your old car - you can sell it privately, but with the paperwork involved and trying to find a buyer, you may be better off trading it in at the dealer. We were told that to trade in the old car, it'd have to be done 48 hours before the new car is given (for a car that is in stock.) Ask different dealers for quotes, though they should be similar. The last car we sold we sold privately, because being an import nobody really gave us a fair price for it. Some of the dealers wanted it for themselves, but the scare of not being able to find parts for imports is palpable... it was also the reason we ended up selling it.