the shake could be from the slow idle being produced by whatever part is going bad. if u have a chunk of belt gone its possibly the alternator belt. the car needs the alternator to run the car after it starts. i lean towards getting the belt replaced. maybe the tensioner is bad? the belt could be going bad, all stretched out or just worn out.

the coil could be a source of the problem if its not the alternator belt. id say the shake is due to not having enough current to run the car or the computer. good luck.