Why does your car shake when you first start it up in the morning or when its cold?

There are many reasons, I would say first, change your oil to something a little thinner. if the shake is for more than a minute, you may want to have a mechanic look at the engine.

thick oil on a cold morning will all settle to the bottom overnight, and the pistons, rods, etc will all rub a bit, and give some vibration. if the oil is coating, it wont do that so much, and thinner oil will be easier on the oil oump in the morning when it is cold, and you should notice a difference.

if it does it longer than a minute, it may be the choke sticking, or a bad o2 sensor, or possibly an airflow sensor. a mechanic can tell you more specifically with the car in front of them, instead of "best guessing" it over the internet. if it only does it when it is cold, maybe leave it at the shop overnight, and meet them in the morning, and start it while they are there and let them see the shake you are describing.