TheShipsList website is one of the "old timers" as web sites go. It has been online since August 1999. This site has over 3,500 totally free access web-pages and still growing. If you are in the U.S., my suggestion is to use this site in tandem with the available ships records on Archive.org to obtain images of the original ship manifests landing in U.S. ports. TheShipsList also has immigration reports, newspaper records, shipwreck information, ship pictures, ship descriptions, shipping-line fleet lists. In addition to US passenger lists you'll also find hundreds of passenger lists to Canada, Australia and even some for South Africa. Word of caution, the search bar at the top does take you to Ancestry.com, a pay site. It's a little time consuming but my suggestion is to either search each individual page or use the Google.com advanced search and let Google limit your search to TheShipsList web site.
[search word or phrase] site:http://www.theshipslist.com