I'm afraid you're probably SOL. Mine does that (skips to the next track for no reason). I first tried re-encoding tracks, then a restore. Also tried using a different format (FAT32 instead of HFS, since I have both Macs and Windows computers available). Nothing.
So, gave it a Google and guess what I found on the Apple Support Forums? A
200-post thread of users with the same problem.
Anyway, to save you reading through the thread, there is some hope: if you set your backlight to 'always on' (and lowest brightness, to save on the battery) it won't skip any more, but it does reduce playtime significantly. If you charge frequently it probably won't matter. Also, if it's within warranty then you might get a replacement, although for some people the replacement had the same problem (and for others, their iPods were returned since nothing was found to be at fault).
Just out of interest, what kind of files do you listen to, and what HDD size Classic is it? Mine are all mp3s (192-320kbps), from a 120GB. I think some people found converting their libraries to AAC solved it, but bleh, hassle. As far as I know there were no other guaranteed suggestions. The last post is actually me expressing my annoyance at some guy who basically said it was just corrupt mp3 files, and that was a bit ago. Wonder if Apple fixed it.
EDIT: And yeah, reset is centre+menu, but I guess you've already figured that one out.