How would I learn more about a sailing vessel and its dates of departure & arrival?

There are several types of records relating to ships including lists of voyages, crew, passengers and cargo.The Maritime History Archive has a great website that provides information on where to locate information on vessels that sailed in Canadian waters.

However, if seeking information on passengers note that these records do not exist prior to 1865.  Passenger lists from 1865-1922 can be found at Library & Archives Canada.

12 year old immigrant's aunt "died on the way over". Would she have been buried at sea, or held for Grossle IIe?

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Does anyone have info on where were the passenger ports in southwest Ontario that took immigrants/emigrants (who had taken a train from Toronto, instead of going through Lake Erie) from Lake Huron to Chicago in the 1880's? For example, were cities like Kincardine, Goderich, or Southhampton passenger ports? What passenger ships departed from these Ontario ports and how long did it take to get to Chicago.

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Where would I find passanger list for various ports?

See the The ShipsList Home Page for information regarding Ships, Passenger Lists and the like.

How do I find out the name of ships that docked at Whitby Harbour during June, 1840?

The best place to start looking for possible records in this instance is by contacting the harbour master at Whitby and asking if log books were kept at that time and if so where they would be located. A lot of harbours have marine and historical societies where one might have access to archives including log books. Contact information for these types of lake front societies may be available at Great Lakes Shipping Genealogy Downward Bound {Special Thank you to Laura Carter for her help in answering this question}

Try the municipal offices of the town of Whitby, or try the Department of Transportation if they have a "Great Lakes" division. Maybe even Archives Ontario under shipping?