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One thing I learned from video editing is that it is not the software that counts, but the monitor. 15 inch is too small, as is 17" or 19". No single screen seems big enough to open all windows you need. If you are serious about video editing, buy yourself a second monitor and a videocard which supports it. Crappy software with two monitors is easier to operate than top-shelf software on a small screen.
For a beginner I would say that the most important thing is to understand the techniques involved, which remain similar whatever software package you use. I would also consider processing power and memory more important than screen size.
So I would say start with serious software (FinalCut on Macs, Avid or Premiere on Windows) if you can, or failing that an entry-level package such as Ulead, and get to grips with the editing tools.
Once you have a command of the basics you can splash out on a 30 inch four million pixel display if you want, but for web video you can probably manage perfectly well without it.