I have used Adobe Premiere because it was bundled with my video capture hardware. It does all that you want and more, although it may be a little too heavy for normal day use.
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.
While it is true that video professionals generally use large screen (and often multiple) monitors I have never had any problem editing video on a 14 inch laptop in FinalCut Pro, which I have been doing since the days of MacOS 9 with excellent results.
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.