It sounds like your alternator belt is loose or worn out. It may be the serpentine belt is worn or loose if one belt drives everything under the hood.
When you start the car, the alternator has to work hard for a while to replenish the energy used to turn over the engine. This makes the alternator harder to turn, which makes the loose or worn belt slip, producing a horrid screeching. Once the battery charge is mostly replenished (and the belt heating up probably helps too), the belt stops slipping and the noise quiets down.
Get it fixed soon--if the belt is worn, it may snap, leaving you stranded. Or if it's good, but just loose, the slipping is wearing it out quickly.