Fo3nix's suggestion of using Keepvid is a good one but you'll also need to convert the file to a suitable format before you can upload it to your phone.
Youtube videos use the Flash Video format. Keepvid doesn't add the relevant file extension to videos. You should rename each file and add .flv to the end of the file name. (e.g. change 'get_vid' to 'mymovie.flv').
You still won't be able to view the movie on your PC (because Windows Media Player doesn't support FLV files) or upload it your phone. To view the stored video on your PC, get either VLC or GOM. (They're both free video players, which support FLV):
VLC:
http://www.videolan.org/
GOM:
http://www.gomplayer.com/main.html
To convert the file to a suitable format for your phone, read the phone's manual to see which file types are supported. Then search for a suitable file conversion utility on either download.com or Tucows:
http://www.download.com/
http://www.tucows.com/
Chris