I have a 1996 mustang V-6 3.8L. Every time I shift it makes a very loud screeching noise and tends to quiet down around 2000 RPMs. The clutch is very stiff when I push down on it. I'm not sure what the exact problem is

 

possibly your throw out bearing

 

First, make sure you clutch is adjusted (lift the pedal up and then slowly press down a few times until you don't hear any more clicking). If that's adjusted right, then your throwout bearing is probably toast. If you're going down there, it's a good time to replace everything (clutch, flywheel, pilot bearing, bearing retainer, etc.).

 

It feels really stiff when I press on the clutch pedal. And it started smelling a little funny. I haven't driven it since the smell. How much would it cost to replace the throw out bearing thru a mechanic? And how much would it cost for that other stuff?

 

Have you tried adjusting it like I suggested?

As I said, if you're going to replace the throwout bearing, you might as well do a full clutch job since you're down there. For parts, you're looking at:
-Clutch kit (which usually includes the throwout bearing). I suggest the Valeo/Sach kit as that's the OEM for Ford. Any other brand of throwout bearing, besides Sach/Valeo/Ford, don't use.
-Flywheel.
-Clutch fork and pivot ball (might be dealer only).
-Pilot bearing.
-Clutch cable.
-Bearing retainer (in case the bad bearing wore into it).

None of these parts are worth it to go cheap on. Either get Ford/Motorcraft or their OEM.

Don't know what the parts will cost, but you can take a look at Rock Auto. As for a mechanic, probably $500-800 would be a good guess. You might get cheaper since the V6/T5 combo is much easier to do then the 4.6/T45/TR3650 combo.