It sounds like you have a drag in the front end accessory drive (FEAD) system. Generally screeching noises occur when the FEAD belt slips.
--Could be a dirty or worn belt. You said the car has 97K miles on it, you said "The belts look pretty new, no cracking." Is that a new belt or is that the original belt with 97K miles on it? If the belt is dirty or worn, it will slip when cold (or when wet, like during a rainstorm). After it slips for awhile, the belt gets hot, and when its hot it gets softer and doesn't slip anymore. The thing to do is to remove the belt, turn in inside out and inspect everything for cleanliness, embedded metal chips, micro-cracks and embedded oil. Clean the belt, and re-install it. Its pretty easy to remove, you may need a breaker bar to back off the spring on the tensioner.
--It could be an unusually large load at startup. This could be the A/C compressor. The way to tell if the compressor is engaged is to take a flashlite, focus it on the front plate of the compressor and see it it spins when the engine is running. That plate should have an air gap between it and the pulley. Measure the gap with feeler gauges (or sheets of paper) when the ENGINE IS OFF. If that air gap is too small, the plate will drag on the pulley and cause a screech noise. If the air gap is too big and the compressor clutch magnet is engaged, the plate will slip, also causing a screech noise. The test for A/C is not to sit in the drivers seat, its to stand to the side of the engine compartment with a flashlite carefully looking at the A/C clutch drive plate at the front of engine (and yes that test is run in park, engine running.. BE SAFE, don't put your hands or body anywhere near that running belt or cooling fan.)
--It could be a large startup load to the alternator. How are the batteries in this car doing? If the batteries are nearly dead, the alternator will be pumping lots of power to recharge them.
--It could be a large startup load to the water pump. When you have removed the belt for cleaning (engine off!), I'd spin the water pump by hand see if the pulley feels firm, no wobble, and just a small load during hand spin. If something feels amiss, that could be an issue (wobble = loose bearings... excess play = impeller hitting the block, etc...) (Note: I doubt its this, but you should check... generally if a water pump goes bad, the noise doesn't get better after a few minutes... it just stays consistent...)
--finally, it could be bad bearings on one of the pulleys.. while the belt is removed (engine off), spin all the pulleys by hand, looking for any unusual free play or pulley/ bearing drag (particularly the idler pulley and the A/C compressor pulley.. with the A/C disengaged, the pulley should spin freely without wobble.)
One suggestion: the purchase of a $30 USD Chilton / Haynes paper manual might be a good investment in the car. It will definitely show you how to remove and re-install the FEAD belt. Removal of the belt should be VERY easy to do. Use of the manual will help you reduce the cost of future expensive repairs if you are remotely do-it-yourself inclined.