Glad you asked and I'll be happy to share what I know:
First, you need two programs: DVDShrink and VirtualDubMod. Both are free and should be easy enough to find. You'll be using DVDShrink first. With the DVD in the drive, open that program, click "Open Disc", and wait for the program to scan the disk. Then click "Re-author" and drag the "title", or section of the DVD with the thing you want to capture, from the box on the right to the box on the left. Right click the title and select "Set Start/End Frames..." and set where on the title you want to capture. Once that's done, click "Backup!" and save the files whereever you want. It should produce two folders. The file you want is in the "VIDEO_TS" folder, and it's a .VOB file.
Now, on to VirtualDubMod. Open the VOB with VirtualDub. Edit the file down to just what you want by clipping out the beginning and end of the clip (DVDShrink is inexact and you'll always have a little bit of unnecessary business at the beginning and end of your VOB files). Once that's done, set your resolution. Do this by going to the "Video" menu and clicking "Filters..." and "Add..." the "resize" filter. For TV shows/pan-and-scan/4:3, resize it to 640 width and 480 height. You will most likely have to deinterlace 4:3 VOB clips to get rid of those unsightly lines. Do that by adding the deinterlace filter. For widescreen/16:9 VOB clips, resize to 853 width (might have to be 852) by 480 height. Filter mode should always be Lanczos3. When it's resized (and deinterlaced if necessary), go back to the Video menu and select "Compression...". Select the Xvid codec if you have it, download and install it if you don't. Xvid is the codec I always use. Once that's all done, save the video.
That's a really basic overview. There's probably a much easier way to do this, but this is the method I've always used. If you need me to go into more detail or step by step, I'd be happy to. Can't wait to see the videos you'll make!
Fango