Car Engine Shaking problem while in first gear at a stop light?

It sounds like your clutch disc is warped. The warpage is probably causing a slight drag in gear that is not there in neutral. Does it shake in any other gear besides first? Your engine/ transaxle mounts could also be at fault as well as rubber subframe mounts.

Most of the answers so far "might" be possible, but I would start by checking for vacuum leaks, & weak sparkplugs, (like wrong gaps or excessive wear) , If you think it might be water in carb/fuel system, the simplest way to "cure the problem" is to go down to the "dollar store" and buy a couple quarts of alcohol and pour it in the gas tank (more than half full)... Alcohol mixes with water, will pull it on through with the gas and burn it, -"no more problem"! When you push down the clutch, it uses a fractional additional amount of power to trun the throwout bearing (especially if the clutch adjustment isn't just right", - when dngine idling, - a small amount of power loss may be enough to make the engine run rough, (or possibly "run light" on one cylinder), - this would be more obvious if it has a vaccuum leak -- since a "light mixture" of fuel to one cylinder woud make weak combustion pulse. If the leak is small,- it woudl naturally "overpower" the dieficiency as soon as the engine speeded up,-- because a whole lot more gas is going in (while the leakage of vaccuum is about the same, - possibly less, because accelerating engine has less vaccuum to begin with)!

Another thing to keep in mind is that more gears are turning in the oil with transmission in gear than in neutral, - oil has a drag "effect" also. On a really cold (sub zero) temperature enviroment, it may be so "stiff' that it requires addition of gas pedal pressure just to keep the engine running - even in neutral! So observe if it is different "in gear" when cold, as opposed to when hot (from driving a while)!

Sounds like the clutch may not be fully releasing. Could be a couple of different reasons for this. It's possible the fingers on the clutch pressure plate are damaged or warped causing the clutch to remain partially engaged, lugging the engine.

Another possibility is a leak in the clutch master/slave assembly which would cause the same condition of a partially applied clutch.

An inspection of the clutch master/slave assembly would be a good idea before condemning the clutch. A leak wasn't all that uncommon on earlier models after they got a few miles on them.

Maybe water in the engine. My car had a similar problem. And it was often difficult to accelerate. If you ever start up your car and get a lot of smoke coming out the exhaust on that initial start-up, it's probably water in the engine.

check your engine mounts....if the rubber on the inside of them is worn out the whole fuggin car will shake...u can buy replacements at a junk yard for like 15 bucks and its only 2 bolts u gotta take off, one in the front one in the back

clutch not completely disengaging from flywheel. could be bad spring in clutch, leak in hydraulics's or self adjuster not working or at it's limits. get repaired soon or you could get stuck in or out of gear while driving and neither is good.