My engine and whole car shakes

First thing I'd check for would be broken motor mount(s). This may or may not be obvious with the engine not running. If you can check them with the engine running, you might be able to tell. Or, you could jack up the engine just a little near each mount to see if they separate. Good luck.

@margaretde57016 have your OBDII port scanned. Because of this "whole car shakes and vibrates when started" I suggest that this could be an issues with the airflow, fuel mixture or ignition modules. Take a look at your spark plugs. See how efficient you burn your fuel.