The British Library has a searchable on-line newspaper archive
Lloyd's List is the newspaper connected with the insurance market, Lloyd's of London. Entries follow the same pattern for most shipping entries in newspapers. These are: Ship's name, master's name and next or final port of call depending on the newspaper. Lloyd's List always gives the next port of call so it can be tracked throughout its journey. Copies of Lloyd's List are held at the Guildhall Library, Corporation of London in the Printed Books section.
Information about masters (here called captains) are also available in the Manuscript Department of the Guildhall Library in the Lloyd's Captains Register collection
Lloyd's Register of Shipping is a different organisation to Lloyd's List. It is published annually in July for all vessels over 300 tons. Details include ownership, port of registry, flag, and technical details of the ship's build. Research can be done in person, or by the staff in the Information Services Dept for a fee if you are unable to visit.
The site also provides information sheets for other sources