Let's suppose you discover that the YouTube video you want to watch is broken into Parts 1-7.
Your first task is to locate the seven parts - not always easy. With a bit of luck you can copy and paste the part of the title common to all parts into the YouTube search bar (go to http://www.youtube.com/) and that might find them. Sometimes you may find two sets of YT videos created by different people. In that case, click the name of the user who posted Part 1, click "View All" on his channel page, and with some luck you may find all the parts. If that doesn't work, go to the bottom of the YouTube page and click "repeat the search" in the line "Some search results have been omitted that contain duplicates. If you like, you can repeat the search with the omitted results included."
Next is to download them as quickly as possible and view them in an uninterrupted sequence. I use Firefox with the DownloadHelper plugin. Remember, the reason for existence of YouTube and all the other Flash Video hosters is to prevent downloading so you are forced to watch all their crappy ads. So it's better to download all the parts first, then view them at leisure.
DownloadHelper won't start downloading to your hard drive until you stop or sometimes even close the page in which the Flash video is embedded. It is then limited to downloading no more than two parts at a time and queues the rest. Since the maximum YT download speed seems to be 60 Kb/sec, this may be frustrating to those of you with more bandwidth, but here's a way around it.
Pause the most recent part. Let's say it's started downloading Part 1 and Part 2, so you pause Part 2. Immediately, Part 3 will begin downloading. Pause that, so that Part 4 begins downloading. Continue until Part 7 is downloading. Now, resume downloading Parts 2-6 and all 7 will be downloading at once, a considerable saving of time if your bandwidth allows it.
Once they are all downloaded, double-click Part 1 to open it in your video player. I use VideoLAN VLC media player. As soon as it has commenced, pause it, press Ctrl+L to bring up the Playlist, Add the remaining parts to it then press Ctrl+L to close the playlist and resume watching the series.
If you wish to join all the videos to make one single .flv file, Google for "freeware flv joiner" and use one of the products available, such as All Free Video Joiner 4.2.1. There are several others. If you plan doing this it is probably better to select the same degree of resolution, e.g., 360 pixels, shown as [360p], for all segments.