The shaking; well that's pretty normal for a Zetec. Probably half the people here have complained about that at some point. It could be a sign of the motor mounts needing replacement though.
The groaning I have no clue about. What kind of "groaning"? Like there's a heavy load on the engine, or like the engine wants to jump clean out of the bay? Mine makes a sound, dunno if I'd call it groaning, it's more of an annoying squeak than anything.
When you pull up to a light, try putting it in neutral, see if that solves it. Works everytime for me. I do it when she's acting particularly jumpy and the light is particularly long.
*Proud Owner of 1962 Chevy Corvette 427