I have accumulated about 20-30 video files on my computer (.asf .wmv
.mpg).  I'm really not into video production or editing, but these
files take up a lot of space on my hard drive. I'd like to reduce
their size by eliminating the boring stuff and keeping just the
interesting stuff.

Is there any software available (free or cheap) via the internet that
will allow me to view each of my files and either...
(1) copy the stuff I want to keep to a new file, or
(2) delete stuff that I don't want?

I tried doing a google search, but there were over a million results. 
I'm hoping that someone can recommend some software that is SIMPLE to
use. (All I want to do is reduce a 20 minute video to a 1 minute video
by deleting 19 minutes of boring stuff.)

Hi there. I'll try to give you a brief guide to what you could
possibly do. I am assuming that you are running a Windows machine and
viewing the video files are as easy as opening them up in Windows
Media Player. Alternately, you can download Videolan Client
(www.videolan.org), a free media player that plays many different
streams.

For the .asf and .wmv files, any editing would probably have to be
done in Microsoft Windows Media software since they are in Windows
Media proprietary format. Microsoft offers a free encoder tool on
their site (http://www.microsoft.com/windows/windowsmedia/9series/encoder/default.aspx).
The software is fairly intuitive and I think you would do okay by
reading the documentation included with the encoder.

As for .mpg files, I would recommend TMPGENC, a freeware utility for
editing MPG files (http://www.tmpgenc.net/e_main.html). After
downloading it, just run the program and under the File menu, there is
a option for MPEG Tools. Select that and go to the tab labeled Merge &
Cut. Load up the MPG file and you can specify where to splice the
video. A window displaying the video is also provided which aids in
finding segments.

Good luck with that and I wish you all the best! =)

My thanks to both sorrento12-ga and o0d3lta0o-ga for your comments!

I do run Windows XP, and I use Windows Media Player to view my video files.

sorrento12-ga, did I understand you correctly?...
  For .asf and .wmv files, I can use Microsoft's free encoder tool
   http://www.microsoft.com/windows/windowsmedia/9series/encoder/default.aspx).
  For .mpg files, I can use TMPGENC (http://www.tmpgenc.net/e_main.html).

You both recommend Videolan Client (www.videolan.org).  Is this
product strictly for viewing? If it also edits, can it edit all 3
video formats that I have (.asf .wmv .mpg)?

o0d3lta0o-ga, you mention Windows Movie maker. Can it edit all 3 video
formats? If so, where can I find it?

Thanks again for your help!
Rambler.

Windows Movie Maker works perfectly! Thank you, thank you, thank you!!!