So long as your PC's specs are up to snuff, the Telltale games should work fine in Vista. Just make sure you run Windows Update before installing, in case you're missing certain files/updates that will help the games run optimally.
Hit the Road (the original game) is more or less its own thing, although TTG's Season 1 can be thought of as a sequel to it. Culture Shock even implies at several points that Sam & Max's last case was the one from HtR. I'm not entirely clear on what you're asking, though, but I hope this helps! ^^;
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But is this the Hit the Road series (season one?)