Use a Video Editing/Composing Program
If you are importing analog video (from an older tape-based recording, e.g. vhs, vhs-c, or Hi8) you may first need to import your video using software supplied by your capture card or external device, before you open up the editing software as described below.
For most of us the video editing machine that comes with the computer (and its operating system) is more than sufficient. The good news is"
No additional cost to make videos
Learn "non-linear" (computer based rather than the "cutting room" professional work) video editing without additional expense.
Tools offered by these programs for editing and "rendering" (producing) the final video is extensive.
The 2 programs that come with their respective operating systems are:
I-Movie - installed on Macs. More intuitive - easier to use - than that of the Windows program. Default output is Quicktime (mov) file. Includes the ability to produce a streaming version of several output formats - very important for your posting of your video on your web site.
Movie Maker - installed on PCs that run Windows XP, Vista, and 7. Default output is Windows Media. Includes the ability to produce a streaming version of several output formats - very important for your posting of your video on your web site.
Your best use begins with playing / interacting with the program. (Play is often a very undervalued method of learning.) Open it, use the sample videos or input your own.
After editing and producing dozens of videos you may be ready to download a free trial of one of the following. Be aware, these are all more expensive. Listed from least expensive at the top...
Power Director from Cyberlink (Great for editing mpg files)