It sounds like you have one hot wire, one switched wire and one neutral. Either the black or the red is from the switch, you'd need a tester to see which one.
Take out the new outlet and look on the side of it. There is a little tab on the side that connects the top and bottom receptacle, and that must be cut off for the switch to work if you are only switching half of the outlet. Only cut off the hot (brass-colored) side, not the neutral(silver-colored) side. then reconnect the three wires, black on one hot screw, red on the other, and white on the neutral side. Then you will have one part of your receptacle hot all the time and one switched