I live in Titchfield, just outside Southampton, where I also used to live. I am also an ex Merchant Seaman, and am able to in form you of where to go.
1 - Ocean Village, This was known as 'the Old Docks' which is on the east of Southampton, where the Queen Elizabeth and Queen Mary (Cunard Line) used to dock, and where the Titanic and Britanic (White Star) left for New York. The Old Dock were also where the Mauritainia (Cunard) and RMS Andes (Royal Mail Line) docked, its now a Yacht Marina.
2 - Mayflower Park, This is on the West of the City, Where the Union Castle Line ships (Athlone - Cape Town - Stirling and several other 'Mail ships') used to sail from. Now apart from the odd one or two ships, you will only see Container Ships, and from the park you can see the Container dock,
where once was a Dry Dock. These used to be known as the New Docks.
Its easy to find, just go into Southampton, and keep driving south, until you get to the Isle of Whight Ferry, left to Old Docks, right to the New Docks.