The squeal when starting is typically caused by the alternator. When you first start the car the battery voltage is low from just expending energy to turn the starter. This causes the alternator to put out maximum until the battery voltage comes up and the regulator cuts it back. That added load causes the alternator to slip on the belt.
The alternator can put out from 100A to 200A. At 150A, that is 2.2KW or about 3 horse power. That is equivalent to turning a 3 horse power motor at full load. During that time, and as long as the alternator is putting out significant amperage there will be a very high pitched whine. That is the brushes singing in the alternator. It should subside as the battery gets charged and the load decreases. Used to be you could demonstrate that by pulling the field wire off the alternator, cutting its output and the sound would stop.
There is a difference between a squeak and a squeal. A squeaking belt will be caused by the ribs rubbing the edges of the pulley(s) as it settles into the grooves. That is a very rapid, short sequence of squeaks.
A sqealing belt is caused by the belt slipping on the pulley and is a long, solid sound. Each has a different sound and a different cause. If we can learn the language of our vehicle we can pretty well diagnose most of the common problems with either our eyes, ears or our hands, or a combination of both.