Q: Peter Jamison came to the US in 1866. What part of Ireland was he born? What port did he leave from and which port did he arrive? He went to Illinois where relatives lived and later to the Gold Fields in Calif.
A: Ships passenger lists to America from 1820 on are available online. You should be able to find your Peter Jamison in Immigration & Emigration Records on Ancestry.com Be sure you use the wildcard feature Ancestry offers to search for him as his surname can be spelled in a variety of ways. (Jamieson, Jameson, Jemison etc) Once you find him on a ships list you will see the port of departure and arrival noted. I would be cautious using 1866 as the firm arrival year. Be sure to search a few years on either side, as the year of immigration is very often mis-remembered.
Q: Was he naturalized? Peter was born in 1844 and died in 1907 in Red Oak, Iowa.
A: The USA Census records will give you citizenship status and dates of naturalization if it occured. Since Peter died before the 1910 census, you should look at the 1900 census. One of the questions asked on it was " the citizenship status of foreign-born individuals over age twenty-one" This will answer your question about naturalization.
To find out what other questions were asked on other census years, see AllCensusRecords.com If he was naturalized you can hunt for his records. Probably the easiest way to do this is to go to Naturalization Records website and use the records and links there.