The first part needs to come from the dealer.
They can provide a set of estimated milestone dates, including when they expect the car to be put on the boat.
You have to check in periodically to see if the dates changed, but once the car is on the boat, the dealer can give you the name of the ship "vessel" and should also tell you which port it will unload the cars. Tracking the ship isn't exact either as the updates are not near real time. Some postings can be delayed by a week or more.
1) Get name of ship "Vessel Name": You can use Vessel Finder to see the "latest" on the ship. My car came in on the Onyx Arrow -
https://www.vesselfinder.com/vessels/ONYX-ARROW-IMO-9267924-MMSI-311918000 If you look at the current URL, you'll see the ship was last reported in San Diego on Feb 19.
2) Get name of port and check their web site for expected arrivals - My car was shipped to Brunswick Georgia. Their website showed what was in port and what was expected -
Georgia Ports Authority > Port of Brunswick > Vessels In Port. You can also use Vessel Finder to see what ships are in port, but the port authority site will be better.
Once the car is in port, you cannot really get any information other than the estimated date for delivery to dealer and that can be off by a week or two. The port is a black box - input is the Vessel and Car. Output is the car. There are no published process pieces while in port.
Hope that helps