I am assuming the OP's car is a manual transmission, correct? If so, you probably have minor warpage of the pressure plate or flywheel surface, which is what causes the vibrations when you engage the clutch. This is usually caused by overheating the clutch from improper use (probably from the previous owner mentioned.) When you eventually have your clutch changed, make sure to replace the pressure plate and have the flywheel resurfaced if possible.
I don't think a gearbox oil change will help with this, although it certainly would be a good idea to replace the fluid with fresh compatible gearbox oil.

I have a BMW 3 series that does the same thing, but that car is going in 100k miles with the original clutch - it only "shudders" every once in a while - probably weather dependent like you mentioned, I just haven't noticed a pattern.