The NSW State Archives & Records is the equivalent in New South Wales of the Public Record Office, Victoria, and it holds similar collections. As with the PROV, the NSW State Archives & Records has a Collection Search that enables you to search for a person across multiple indexes. Some of the records available on the website are browse only, such as the Assisted Immigrants (digital) Shipping Lists. To find someone on these lists, the best way is to search for the person in the Collection Search, to see if there is an entry for them from the Assisted Immigrants Index 1839-1896. This will then give you the year and the name of the ship, which will make them easier to find on the passenger lists.
I haven’t used this site much for other records yet, so a good place to start when searching the site is to read through the page Using the State Archives Collection
Most of the records are held at the Reading Records in Kingswood. A copying service is available for some of the records for those unable to visit the reading room. Some of the more popular records have been copied to microform, and are available at Community Access Points.