video_ts files are for DVD videos. I assume you have ripped the DVD and lost any protection along the way.
There are many methods to do what you want
http://www.autogk.me.uk/ is able to do simple conversions and is very easy to use.
http://handbrake.fr/details.php is fairly self contained and does not require much thought.
http://forum.doom9.org/forumdisplay.php?f=17 has a bunch more you might want to look at.
If you prefer though (the following method if you know what goes will generate the best quality you can hope for at a good size)
http://neuron2.net/dgmpgdec/dgmpgdec.html will turn the component MPEG2 files and the audio tracks as well as a sort of index of the MPEG2 data.
You can load them with several methods but most opt for avisynth. Avisynth is something of a scripting language but you should only need the basics.
You can then feed them to your chosen encoder (typically XVID or H264), there are programs to make life easier like megui
http://sourceforge.net/projects/megui/
This will make a video file and an audio file (it might merge them but you will probably have to tell it to do so). You want a muxer for this and there are many, you said AVI though so
http://www.alexander-noe.com/video/amg/
Others include avidemux and virtual dub. Megui should have a muxer as well.
After this if you want subtitles then you can either rip them (DVD subs are pictures with timing data) so if you want proper subs you will need to OCR them.
Subtitle edit
Subrip- a bit older but still good
Those programs should also be able to generate vobsub type subs (the idx and sub files you might see when you download films).
I usually flank things with subtitle workshop and subtitle creator or maybe Aegisub- a bit complex but probably the best subtitle program out there at the moment.