ballahboy, those subs look like they are probably .srt format. Try renaming the .txt file to .srt. The .srt file must have exactly the same name as your .avi file before the extension. WinAVI and VisoXtoDVD both have support for .srt files to be added when converting. You can use BSplayer to check if your subtitles are working, just make sure they are in the same folder and the .srt is named exactly the same as the .avi.
David, .sub with and .idx is the VobSub format, and you need both the .sub and the .idx (the .sfv is something different, don't worry about it). You will need to install VobSub. Use BSplayer to make sure the subtitles play and make sure they are named the same as your .avi file. I believe VisoXtoDVD supports these natively, and WinAVI will include them as long as VobSub loads them up and you have the preview window open when you convert (just to check and be on the safe side). I'm not sure, but WinAVI may natively support that format now, I've not checked myself. There are other converters that also handle various sub formats.
Authoring programs like DVDLab, can also allow you to include subtitles after conversion, but although I know of the feature I've not yet used it.