My 1999 Honda Civic makes a loud screech when starting the car. It usually quits with in a few seconds, but is very loud. It does this longer when its cold out.
The accessory drive belt(s) are loose and probably worn. The noise is the belt(s) slipping on their respective pulleys. Considering the age of the car I’d recommend new belts properly installed and adjusted.
Has the timing belt ever been replaced? If not this would be a good time to do it. The timing belt is not making the noise, but it should be replaced to prevent engine damage if it should break.