As sort of a family tradition we have created videos for our children on their birthdays (some years, have a lot to do there) and of holidays and vacations to share with family and friends. It is a fun thing to do if you can get it down to just the good pictures and short video clips.
I started off using the Microsoft Movie Maker software that comes for free with Windows XP, and is a fine place to start. I quickly outgrew Movie Maker, wanting to be able to do some things like panning and cropping where I wanted it to rather than the canned panning and cropping that Movie Maker will do. The software of choice for me is Vegas Movie Studio.
I am using version 8 right now, and for the price ($74.95 as of this post) I don't think you can beat the software. It allows for multiple video and sound tracks, does a great job with transitions, and doesn't get unresponsive if you have more than 10 minutes worth of video in your project (an issue I saw frequently with Movie Maker). The one area it lacks a little for me now is with the ability to add titles. It does a fine job with full screen titles and after video credit rolls, etc, but it takes a lot of work to get a title to display at the top or bottom. It is an app that takes a little getting used to, but well worth the effort
If you are going to do any video editing you are going to need some hardware. I'd say at least a dual core processor, 2GB RAM, and a good video card. These are the kind of specs nearly any PC has these days, but you'd have to make sure a laptop had that much hardware because that is still an "upper" end configuration for a laptop.
I don't have any real experience with Adobe Premier, the other software that is often rated or compared with Vegas Movie Studio. The biggest issue I have with it is the cost. Premier may very well be a better video editing package, but the cost difference between it and Vegas Movie Studio is incredible.