Southhampton to London

Depends..
Picking times the fare can be £8 from Southampton Central station,the slow stopper is 2.30h journey on Southern Railways and. goes to Victoria.
The walk up ticket is generally £36 off peak, valid on all routes the faster services tend to have 1/2 changes and be 1hour 30.
nationalrail is the start point, if you want to prebook or check your prefered options.
I'd be surprised if there isn't a shuttle bus or a queue of taxis at the port when a ship comes in.

Thanks Richard!

Bill, we have done a private transfer from S'hampton to London, with 4 sharing the cost. If you google such a transfer a lot of names will come up. It was lovely to be met on the dock and have our driver handle the luggage.

There are three bus routes and a couple of train runs that depart Southampton for London. At least one bus run and one train run goes to Victoria Station. Here's a link to more information with great route graphics. http://www.londontoolkit.com/travel/southampton_london_transfers.htm