Well, assuming you already know how to capture the video (and audio) onto your computer and into some media file, you can then use a DVD Authoring program to author your DVD. The only software I know to do this is
Adobe Encore. Truth be told, it's slow and buggy as hell but, if you're patient, it eventually works. This happens to be what I use.
Depending on what authoring program you use, you may need to encode the movie (with audio) into MPEG-2, in which case you will need some kind of MPEG encoder program and a familiarity with the DVD format protocol since not just any old MPEG encoding will do. Luckily for Adobe Encore users, it can do the transencoding itself, so you won't need to know any of this stuff and can simply import AVIs or likely whatever else.
The only way to "guarantee" that your DVD will play in any DVD player is to press the DVD (there's always the possibility of a poor player not playing some DVD, like first generation Sony Playstation 2s). There's no way you can afford this kind of equipment so you will have to outsource an order to some company that does this. Modern DVD players will play almost any DVD-R you burn so this shouldn't be a deep concern for you.
I hope this was helpful. If you need any more help or clarification, we're still here...