As another person who used to live in Southampton, I would recommend staying in Southampton and getting the train across for a day trip(s) to Portsmouth. As davidjohnson6 says, Southampton is more a place people go through, while Portsmouth oozes Naval history and the historic dockyard area is a fantastic visit. That said, Southampton is a 'nicer' place to stay (and no I'm not biased basis the rivalry between the two cities
.) The nice area of Southampton for an evening meal is Oxford Street.
If you have time another excursion that is worth looking into is getting the Red Funnel - Red Jet ferry service from Southampton to Cowes on the Isle of Wight. A town more famous for yachting than navy, but has lots of charm and lots of historic pubs with character in its old town, all walkable from the ferry terminal.
Alternately, for the traditional 'country pub' feel then heading to the New Forrest National Park is worth looking into. There are more traditional pubs to choose from in its borders then you will know where to start. You will need to get a taxi from Southampton, plan 20-30 mins depending. If you are lucky you may see a wild pony the area is famous for.