Do I need to convert the files to a different format?
Yes, if you wish to play them on an TV device.
If so, what format
Most people use H.264/AAC/AC3 compressed content. H.264 provides moderately high quality in a relatively small file size. The AAC audio makes the file compatible with QuickTime/mobile media devices. AC3 passthrough allows you to retain DD 5.1 audio compatibility if the TV playback is connected to an appropriate decoder/speaker array.
what converter software should I use
That is up to the individual user. I generally use HandBrake when higher quality is desired and the Turbo H.264HD hardware accelerator/converter when speed is more important than quality.
will the quality still be DVD quality when viewed on a big-screen TV?
That will depend on the settings you use. The H.264 codec can be used with a wide range of data rate settings. Slower settings produce small files with less quality while high data rates should reproduce the original visual quality at the expense of a larger file size. Generally speaking, you can use a data rate about half that of your original MPEG-2 data rate and still maintain a level of quality approximating the original DVD.
Is this going to be ridiculously time-consuming?
It can be depending on the settings and method used. Most people employ a single pass, constant quality conversion strategy and batch process their files. This means the files can be converted automatically while you sleep, are at work, or otherwise not actively using your system. This strategy is most efficient in terms of time and quality while a multi-pass strategy is probably more efficient in terms of file space.
I was hoping that purchasing an Apple TV might be a good solution for me, but if the files won't play in iTunes, then they won't play via Apple TV onto my HDTV, right?
VOB files are not natively supported by either QT or mobile devices. Such files can only be played in QT/iTunes if 1) you have installed the QT MPEG-2 Playback component, have added hardware software or hardware support for AC3 playback, and 3), in the case of iTunes, have placed the content in an MOV file container so it can be imported/managed by iTunes. If you insist on playing the VOB content, a better approach would be to extract the main Title VOB set as a VIDEO_TS folder and use either the DVD Player or Front Row for playback. Unfortunately, this means your computer must be directly connected to your HDTV or projector. In this case, most users would use a dedicated Mini (or similar platform) as the media player.