I have a 1998 Mazda Protege with 160,000 miles on it. 9 out of 10 times when I turn on the engine - it makes a very loud high pitched screeching sound (seems like its coming out of the hood of the car). There are no “check engine” lights on and the car runs fine. The screeching sound lasts about 30 seconds and then poof - it stops. This happens in all kinds of weather (hot, cold). My mechanic couldn’t find anything wrong with the car (and of course it never makes the screeching sound when at the mechanic’s). The noise does not bother me much - but I think my neighbors now hate me 
Is this something serious? or maybe just a loose belt or something? Please advise - keep in mind I know nothing about cars.
If your mechanic can’t find the problem, it may be time to look into a different mechanic. Either that or leave the car at the shop overnight if the noise only happens in the morning or after the car sits for a while. This problem should be easy to diagnose. It could even be a simple case of worn belts. If they’re old, even if they seem tight enough, they can still slip and make noise.
Wore out belts or belts that are too loose/slipping can create this sound, I know its unpleasant. Your car needs to spend the night with your mechanic. Bring him the key in the morning and have him start it up and listen. Like Mark said, shouldn’t be hard to diagnose. Oh check out the car talk “noise emporium” and listen to the loose belt sound, see if it matches up.
I agree with the previous posts. A mechanic that can’t find a bad fan belt is probably not a mechanic. With the engine off…just pull on the drive belt(s) and twist them over at the same time. You should hear a nasty noise if they are bad.
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.