Not too sure about places to eat in that area but you should not have any issues just asking around. There is just about anything to be found in London.
As far as getting around, I would look into the train service, then cab/bus from the station. Most any place in the UK is serviced by a train and I'm pretty sure you'll have a number of options from London to Southampton. You might find some stations are easier than others for getting around with luggage so check into them first. The best rule of thumb for tube/train travel in the UK is only take what you can manage with your two hands. While most all stations have an elevator/escalator they are sometimes out of the way.
Here is one link but definitely shop it to get the best deal. Also check with Princess, they might have a bus/connection from the station available. http://www.directrail.com/london_southampton_train.html<</a>/p>
There are so many, you will have no problem finding inexpensive places.
For one meal, at one inexpensive location, I would really recommend The Audley. Close to the American Embassy on Audley street (Behind The Marriott Grosvenor) you can enjoy a great meal. Many Americans will be there and it is a great experience.
Note the pictures! Have a great time.
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