There's lots of low-cost video editing software available today. But software is just a tool. And before you start concentrating on which tool to buy, you should learn the process first. Believe it or not, you probably already have software that'll help you do this for free.
Windows XP computers come equipped with free editing software called Windows Movie Maker (the software is also available for download). The latest Mac machines are loaded with iMovie. Both of these software packages are designed to work with DV video that you feed into your computer through your FireWire or iLink port. (By the way, FireWire and iLink are just marketing names - they refer to the DV connection on your computer.)
These are basic software packages, so you won't be able to do a whole lot of video and audio layering. But, for free software, each is surprisingly good - certainly good enough to help you learn the basics of video editing.
After you've learned the basics, you may find that your free software meets your needs. In this case, you certainly shouldn't spend money on additional software. If, however, your free software limits you creatively, it's time to go shopping.