Hi Nancy, and welcome to the forum.
If you are only doing minor editing, you can edit directly in MPEG format. Some MPEG editors are: Womble Mpeg and TMPGenc MPEG, both of which are reasonably priced. There are some free ones as well, but I'm not all that familiar with them. With you using a DVD Recorder, your original output will be MPEG format. I also recommend that you use quality media (DVD's), such as Verbatim, Taiyo Yuden, FujiFilm (Made In Japan). Most other media's are a crap shoot, unless you research which firmwares and write strategies work well with that particular media. A DVD that will burn good in one drive, may be junk in another drive. That doesn't happen with the media I suggested.
If you are doing more than minor editing, it is best to demux the video into avi format. The list of tools for doing that is quite long. I don't do that, so I can't really recommend a tool for that. Others here will have good input, and you can find a lot of info at www.videohelp.com.
If you are new to video and video editing, I really suggest that before you get too involved, spend some time reading some of the information posted in the forums at www.videohelp.com. This will give you a good grasp of what you need to do, tools, and information on how to maintain high quality video.
If you have questions after doing the reading at videohelp.com, feel free to come back here and ask questions, or you can also ask questions at the videohelp forum.