What causes my computer to make grinding noises right when it is opening
up all the programs at the end of the start process?  It sounds like
the noise a car makes when you keep turning the key and it does not start. 
After a few minutes, it opens up and works just fine, but rather
slowly.  When I shut down the computer, it makes the same sound but
only for a few seconds.  I have tried defraging, scanning for viruses
and damage (there were none) and I do not have a firewall or any
spyware installed.

Maybe something here will help...

Coping - computer noises
http://www.haruteq.com/cope-04.htm

Good luck,
hummer

My guess is that a non-essential mechanical object is loose or
misaligned in a hard drive. Your system still works so I say it is
non-essential. The noisy periods seem to me to coincide with a high
level of hard drive activity.

Don't be shy to run your PC with the case open and try to localize the
sound within the box.

Could also be a dodgy fan - say on the processor

As Augusta said, open the box and listen
- using a bit of flexible tube as a 'stethoscope' also helps

If you find the noise is coming from a disk unit then replace it - fast

Oh yeah, maybe it's a fan. When the plastic fan warms up enough and /
or spins fast enough it quiets down. Then as the PC is shutting off
and it slows down it gets noise again.

This is theory #2.

Do u own a dell? anyways, the problem is usually indicative of a worn
bearing on a secondary internal unit cooling fan; my suggestion is to
take the cover off the tower (depending on design) and simply with a
plastic pen (bic is good) force each fan to stop rotating until you
find the defective one. This will not cause any damage to your system,
as long as you don't force the pen into the fan and do it for just a
few seconds to see if the grinding stops when you find the suspect
fan. It's most likely not the processor fan as it would be a perennial
grinding not intermittent. therefore i'm confident it's one (depending
on design) of your cooling fans. When you find suspect fan, simply
remove it and take it to most computer stores to procure a
replacement. I would suggest doing it soon as a delay could lead to
future havoc especially if it happened to be the processor fan.

Do you use your Floppy Disk Drive often?  If this is broken or has
something stuck in it a program could be accessing the drive on
startup and also on shutdown.  Try reading a NON essential disk and
see if it makes the same noise.  If so just replace it, there's no
point trying to fix something as cheap as this.

Hi,

I would suggest you backup all your data on your hard drive/s , there
are some symptoms of hard drives that makes such noises before they
actually fail. While you still have the chance, try to save as much
data as you can before it's too late.

Normally if the bootup is too slow and there's a grinding noise It is
a hard drive that's about too fail.

Regards,
168c