I have flown into Plymouth a few times, but not to stay for very long... Tip. You will probably have to park on the sloping apron, so USE THE CHOCKS provided. (okay, granny and eggs and all that, but worth a reminder)
However, I did grow up there so still know a little about the place.
Things to do? Take a taxi down to
The Barbican. (Will only take about 15 min on a good day). This is the Old part of Plymouth. The streets still have original cobbles. You can see the "Mayflower Steps", where The Pilgrims finally left England for the New world. There are many Elizabethan Houses. (look at the small Windows and doors).
Also here is a plaque commemorating the first crossing of the Atlantic by Air. Achived by the USN with several Curtis Flying boats, and lots of warships stung out across the pond. Can't remember dates, but I'm sure someone will advise.
The National Marine Aquarium is a short walk away
http://www.national-aquarium.co.uk/about/
Or, you can walk up onto The Hoe, for a magnificent view of Plymouth Sound and Drake's Island.
Food. Gotta be a Pasty, if you're in Plymuff.
Enjoy
Rgds BEX