TIMELINE pre1855- Immigration is the responsibility of the state. 1855- Castle Garden, NY is used for the influx of immigrants. 1890- Castle Garden ceases processing immigrants. 1890- The U.S. Government selects Ellis Island as the Federal Immigration Center for New York. 1892- The Ellis Island Immigration Center opens. 1897- Fire destroys the original buildings on Ellis Island. 1900- Ellis Island Immigration Center re-opens, in the building that still exists to this day. 1924- The National Origins Act dramatically reduces immigration flow. 1954- Ellis Island Immigration Center closes.
There were other immigration points that probably continued functioning into the 1860s, but you may find that persons destined for elsewhere actually "landed" at New York before proceeding to their final destination. Few records were kept by the immigration people, but the do have ship manifests that at least list departure points and dates, arrival dates, and lists of passengers with some description for identification.
There is a brief history of immigration specifically at Ellis Island that may give useful background - at least by way of suggesting how other ports of immigrant entry may have worked in the era of interest.
The site also has a link for a "free trial" access to Ancestry.com (14 days). Ancestry.com is by far the largest single search source for US ancestral records but ... Be aware that this is a "one time only" free period, not all records that they have are accessible without additional fees, and "subscribing" for regular use after the free trial is extremely expensive. (i.e. You may be able to find quite a lot, but plan your search to the extent possible before you sign up, and do it when you have time to follow as many leads as possible during your two week trial.)
A Google search on the city of arrival + "immigrant" may find similar records for other ports(?).