So I have two separate, but seemingly related problems right now with my 1996 Honda Civic. Pretty much, it squeaks...

The first, I believe I know what is wrong. Whenever you press the power window buttons, or switch on the headlights, etc... there is a faint squeaking that goes on (until you release the button, etc)... I had a "mechanic" look at it, and he saw that the belt connected to the alternator didn't have enough tension... I trust his judgement, I just don't know how to fix it.

Second, when I start the car up when its cold, it makes a very loud screech... my friend heard it across a lake. If I let it run for 30 sec - 1 min, turn off engine,start it again, usually it doesn't screech anymore. Also, when I start it up warm, no screech. USUALLY... SO, this is really embarrassing and I want to do all I can to fix it.

Your mechanic should have adjusted the tension on the belts while he had his interest in your engine compartment. Schreeching from the belts indicates over or undertightened belts or even worn belts/worn alternator (or pulley driven pumps) bearings. When you demand power from the elctrical system (i.e. headlights/ power windows)you are putting a load on the alternator, which makes it work harder. Have an experienced tech adjust the belt tension(or replace it, if it has not yet been replaced) as well as inspect your alternator. Typically the reason the screeching fades after engine warming is because metal expands when heated(engine pully) thus correcting tension on the belts.